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.     

Quasistatic propagation of steps along a phase boundary

We study quasistatic propagation of steps along a phase boundary in a two-dimensional lattice model of martensitic phase transitions. For analytical simplicity, the formulation is restricted to antiplane shear deformation of a cubic lattice with bi-stable interactions along one component of shear strain and harmonic interactions along the other. Energy landscapes connecting equilibrium configurations with periodic and non-periodic arrangements of steps are constructed, and the energy barriers separating metastable states are calculated. We show that a sequential one-by-one step propagation along a phase boundary requires smaller energy barriers than simultaneous motion of several steps.      

薄膜涂层材料中的广义瑞利波

基于弹性动力学的线性理论，建立了涂层材料中广义瑞利波传播的理论分析模型，并 且由波动方程和边界条件推导了波的频散方程．分析了慢层和快层对相速度频散的影响，给 出了不同层厚-波长比和不同涂层-基体密度比情况下广义瑞利波相速度的理论解．算例分 析分别比较了慢层和快层结构中波的相速度、群速度，以及随深度衰减的位移与应力振 幅．另外，相速度曲线和位移振幅曲线与文献中给出的结果吻合，验证了理论模型和分析过 程的正确性．     

Towards a transparent boundary condition for compressible Navier–Stokes equations

A new artificial boundary condition for two‐dimensional subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow, as well as a convection velocity, is used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems, whose solutions are known, and comparisons with other boundary conditions (BCs), show the efficiency of the BC. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow. Copyright © 2001 John Wiley & Sons, Ltd.     

Interaction between an unsteady pressure disturbance and a hypersonic boundary layer

The development of disturbances in a hypersonic boundary layer on a cooled surface is investigated in the case in which the characteristic velocity of disturbance propagation is small but greater than the flow velocity in the wall region of the three-layer disturbed zone with interaction. The nonlinear boundary value problem formulated involves a single similarity parameter that characterizes the contribution made by the main, on average either subsonic or supersonic, region of the boundary layer to the generation of the pressure disturbance. In the linear approximation, an analytical solution and an algebraic dispersion equation are derived. It is shown that only waves exponential in time and in the streamwise coordinate can propagate downstream when themain region of the undisturbed boundary layer is subsonic on average.     

Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models

We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid–solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of the existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory.     

An analysis of dislocation nucleation near a free surface

Molecular dynamics analyses of defect-free aluminum single crystals subject to bending are carried out to investigate dislocation nucleation from free surfaces. A principal aim of the analyses is to provide background for the development of dislocation nucleation criteria for use in discrete dislocation plasticity calculations. The molecular dynamics simulations use an embedded atom potential for aluminum. Bending is imposed on a strip by specifying a linear variation of displacement rate on opposite edges. The overall bending response is determined and the character of the dislocations nucleated is identified. It is found that the stress magnitudes at the instant of dislocation nucleation are nearly an order of magnitude smaller than for homogeneous bulk dislocation nucleation. The characterization of dislocation nucleation in terms of various phenomenological nucleation criteria is explored, in particular: (i) a critical resolved shear stress; (ii) the onset of an elastic instability; and (iii) a critical stress-gradient criterion. It is found that dislocation nucleation is not well-represented by a critical value of the resolved shear stress but is reasonably well-represented by the critical stress-gradient criterion.     

Saint-Venant decay rates for an anisotropic and non-homogeneous mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects in anti-plane shear deformations of linear mixtures of elastic solids. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear anisotropic mixtures of elastic solids. As our first results have a very general point of view, we study some examples in detail.     

Bleustein–Gulyaev waves in 6 mm piezoelectric materials loaded with a viscous liquid layer of finite thickness

In this paper, we analyze the propagation of Bleustein–Gulyaev waves in an unbounded piezoelectric half-space loaded with a viscous liquid layer of finite thickness within the linear elastic theories. Exact solutions of the phase velocity equations are obtained in the cases of both electrically open circuit and short circuit by solving the equilibrium equations of piezoelectric materials and the diffusion equation of viscous liquid. A PZT-5H/Glycerin system is selected to perform the numerical calculation. The results show that the mass density and the viscous coefficient have different effects on the propagation attenuation and phase velocity under different electrical boundary conditions. In particular, the penetration depth of the waves is of the same order as the wavelength in the case of electrically short circuit. These effects can be used to manipulate the behavior of the waves and have implications in the application of acoustic wave devices.     

Analysis of two-dimensional,finite amplitude wave propagation by time marching methods

Two-dimensional, finite-amplitude wave propagation in an inviscid, subsonic, perfect gas medium is analysed by explicit finite-difference methods. A two-step, Lax-Wendroff method and the single-step, Lax-Friedrichs method are used. A prescribed propagating velocity or pressure disturbance is applied along a single row of grid points normal to the stream direction and results in a 'forced' outflow boundary. The inflow boundary is placed far from outflow by utilizing a streamwise expanding grid and uniform inflow is imposed. Side boundaries are spatially periodic. The numerical solutions are compared with analytical small-perturbation solutions; higher-order effects arising from non-linearities are revealed by Fourier analysis. Solutions which closely approached a periodic state were obtained. The Lax-Wendroff method combined with the expanding grid is shown to be accurate and stable, the Lax-Friedrichs scheme produced highly damped solutions.     

Solitary waves in a peridynamic elastic solid

The propagation of large amplitude nonlinear waves in a peridynamic solid is analyzed. With an elastic material model that hardens in compression, sufficiently large wave pulses propagate as solitary waves whose velocity can far exceed the linear wave speed. In spite of their large velocity and amplitude, these waves leave the material they pass through with no net change in velocity and stress. They are nondissipative and nondispersive, and they travel unchanged over large distances. An approximate solution for solitary waves is derived that reproduces the main features of these waves observed in computational simulations. It is demonstrated by numerical studies that the waves interact only weakly with each other when they collide. Wavetrains composed of many non-interacting solitary waves are found to form and propagate under certain boundary and initial conditions.     

A failure criterion based on material instability

Scaling laws for adiabatic shear bands are used to parameterize a model that is suitable for introducing shear damage within engineering calculations. One-dimensional solutions to the governing equations for a single shear band provide laws that connect the driving deformation, the imperfections, and the physical characteristics of the material to the process of stress collapse [International Journal of Plasticity 8 (1992) 583, Mechanics of Materials 17 (1994) 215]. The current model uses homogeneous material response and the scaling laws to anticipate the correct timing beyond the maximum stress at which stress collapse should occur. The model is implemented into a finite element code for wave propagation and used in the analysis of boundary value problems that are dominated by shear failure. Finally, implications of the model for simulations of material failure are discussed.     

Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper

Nanoscale contact of material surfaces provides an opportunity to explore and better understand the elastic limit and incipient plasticity in crystals. Homogeneous nucleation of a dislocation beneath a nanoindenter is a strain localization event triggered by elastic instability of the perfect crystal at finite strain. The finite element calculation, with a hyperelastic constitutive relation based on an interatomic potential, is employed as an efficient method to characterize such instability. This implementation facilitates the study of dislocation nucleation at length scales that are large compared to atomic dimensions, while remaining faithful to the nonlinear interatomic interactions. An instability criterion based on bifurcation analysis is incorporated into the finite element calculation to predict homogeneous dislocation nucleation. This criterion is superior to that based on the critical resolved shear stress in terms of its accuracy of prediction for both the nucleation site and the slip character of the defect. Finite element calculations of nanoindentation of single crystal copper by a cylindrical indenter and predictions of dislocation nucleation are validated by comparing with direct molecular dynamics simulations governed by the same interatomic potential. Analytic 2D and 3D linear elasticity solutions based on the Stroh formalism are used to benchmark the finite element results. The critical configuration of homogeneous dislocation nucleation under a spherical indenter is quantified with full 3D finite element calculations. The prediction of the nucleation site and slip character is verified by direct molecular dynamics simulations. The critical stress state at the nucleation site obtained from the interatomic potential is in quantitative agreement with ab initio density functional theory calculation.     

On criteria for dynamic adiabatic shear band propagation

In this work, we postulate the physical criterion for dynamic shear band propagation, and based on this assumption, we implement a numerical algorithm and a computation criterion to simulate initiation and propagation of dynamic adiabatic shear bands (ASBs). The physical criterion is based on the hypothesis that material inside the shear band region undergoes a dynamic recrystallization process during deformation under high temperature and high strain-rate conditions. In addition to providing a new perspective to the physics of the adiabatic shearbanding process and identifying material properties that play a crucial role in defining the material's susceptibility to ASBs, the proposed criterion is instrumental in numerical simulations of the propagation of ASBs when multi-physics models are adopted to describe and predict the complex constitutive behavior of ASBs in ductile materials. Systematic and large scale meshfree simulations have been conducted to test and validate the proposed criterion by examining the formation, propagation, and post-bifurcation behaviors of ASBs in two materials, 4340 steel and OFHC copper. The effects of heat conduction, in particular the length scale introduced by heat conduction, are also studied. The results of the numerical simulations are compared with experimental observations and a close agreement is found for various characteristic features of ASBs, such as the shear band width, speed of propagation, and maximum temperature.     

A fully hydrodynamic model for three‐dimensional,free‐surface flows

The hydrostatic pressure assumption has been widely used in studying water movements in rivers, lakes, estuaries, and oceans. While this assumption is valid in many cases and has been successfully used in numerous studies, there are many cases where this assumption is questionable. This paper presents a three‐dimensional, hydrodynamic model for free‐surface flows without using the hydrostatic pressure assumption. The model includes two predictor–corrector steps. In the first predictor–corrector step, the model uses hydrostatic pressure at the previous time step as an initial estimate of the total pressure field at the new time step. Based on the estimated pressure field, an intermediate velocity field is calculated, which is then corrected by adding the non‐hydrostatic component of the pressure to the estimated pressure field. A Poisson equation for non‐hydrostatic pressure is solved before the second intermediate velocity field is calculated. The final velocity field is found after the free surface at the new time step is computed by solving a free‐surface correction equation. The numerical method was validated with several analytical solutions and laboratory experiments. Model results agree reasonably well with analytical solutions and laboratory results. Model simulations suggest that the numerical method presented is suitable for fully hydrodynamic simulations of three‐dimensional, free‐surface flows. Copyright © 2003 John Wiley & Sons, Ltd.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

Time-dependent plane Poiseuille flow of a Johnson–Segalman fluid

We numerically solve the time-dependent planar Poiseuille flow of a Johnson–Segalman fluid with added Newtonian viscosity. We consider the case where the shear stress/shear rate curve exhibits a maximum and a minimum at steady state. Beyond a critical volumetric flow rate, there exist infinite piecewise smooth solutions, in addition to the standard smooth one for the velocity. The corresponding stress components are characterized by jump discontinuities, the number of which may be more than one. Beyond a second critical volumetric flow rate, no smooth solutions exist. In agreement with linear stability analysis, the numerical calculations show that the steady-state solutions are unstable only if a part of the velocity profile corresponds to the negative-slope regime of the standard steady-state shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to different stable steady states, depending on the initial perturbation. The asymptotic steady-state velocity solution obtained in start-up flow is smooth for volumetric flow rates less than the second critical value and piecewise smooth with only one kink otherwise. No selection mechanism was observed either for the final shear stress at the wall or for the location of the kink. No periodic solutions have been found for values of the dimensionless solvent viscosity as low as 0.01.     

Dynamic interfacial crack propagation in elastic–piezoelectric bi-materials subjected to uniformly distributed loading

In transversely isotropic elastic solids, there is no surface wave for anti-plane deformation. However, for certain orientations of piezoelectric materials, a surface wave propagating along the free surface (interface) will occur and is called the Bleustein–Gulyaev (Maerfeld–Tournois) wave. The existence of the surface wave strongly influences the crack propagation event. The nature of anti-plane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids. Piezoelectric surface wave phenomena are clearly seen to be critical to the behavior of the moving crack. In this paper, the problem of dynamic interfacial crack propagation in elastic–piezoelectric bi-materials subjected to uniformly distributed dynamic anti-plane loadings on crack faces is studied. Four situations for different combination of shear wave velocity and the existence of MT surface wave are discussed to completely analyze this problem. The mixed boundary value problem is solved by transform methods together with the Wiener–Hopf and Cagniard–de Hoop techniques. The analytical results of the transient full-field solutions and the dynamic stress intensity factor for the interfacial crack propagation problem are obtained in explicit forms. The numerical results based on analytical solutions are evaluated and are discussed in detail.     

Shear band propagation from a crack tip subjected to Mode II shear wave loading

Results are reported for pressure–shear plate impact experiments in which pre-cracked 4340 steel plates are subjected to Mode II loading. Experiments show the propagation of a shear band ahead of the initial crack. Finite element simulations are used to interpret the results. Normal and transverse velocity–time profiles measured at the rear surface of the target can be simulated reasonably well using even an elastic model for the material response. A propagating shear band is obtained when the material is modeled as having reduced shearing resistance described by a thermo-viscoplastic power law, and complete loss of shearing resistance when the shear strain reaches a critical value. However, the predicted speed of propagation of the tip of the shear band is substantially less than required to explain the lengths of the bands observed in the experiments. Adjustments of parameters of the power-law model have little effect on the overall length of the band. Possible reasons for differences between predicted and measured shear band speeds are examined. Further reduction in the shearing resistance in the shear band appears to be essential for the simulated bands to be as long as those observed in the experiments.