A single theory for some quasi-static,supersonic, atomic,and tectonic scale applications of dislocations

We describe a model based on continuum mechanics that reduces the study of a significant class of problems of discrete dislocation dynamics to questions of the modern theory of continuum plasticity. As applications, we explore the questions of the existence of a Peierls stress in a continuum theory, dislocation annihilation, dislocation dissociation, finite-speed-of-propagation effects of elastic waves vis-a-vis dynamic dislocation fields, supersonic dislocation motion, and short-slip duration in rupture dynamics.     

Nucleation of partial dislocations at a crack and its implication on deformation mechanisms of nanostructured metals

Nucleation of partial dislocations at a crack is analyzed based a multiscale model that incorporates atomic information into continuum-mechanics approach. The crack and the slip plane as the extension of the crack are modeled as a surface of displacement discontinuities embedded in an elastic medium. The atomic potential between the adjacent atomic layers along the slip plane is assumed to be the generalized stacking fault energy, which is obtained based on atomic calculations. The relative displacements along the slip plane, corresponding to the configurations of partial dislocations and stacking faults, are solved through the variational boundary integral method. The energetics of partial dislocation nucleation at the crack in FCC metals Al and Cu are comparatively studied for their distinctive difference in the intrinsic stacking fault energy. Compared with nucleation of perfect dislocations in previous studies, several new features have emerged. Among them, the critical stress and activation energy for nucleation of partial dislocations are markedly lowered. Depending on the value of stacking fault energy and crack configuration, the saddle-point configurations of partial dislocations can be vastly different in terms of the nucleation sequence and the size of the stacking fault. These findings have significant implication for identifying the fundamental dislocation and grain-boundary-mediated deformation mechanisms, which may account for the limiting strength and the high strain rate sensitivity of nanostructured metals.     

A finite-deformation,gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations

This paper develops a finite-deformation, gradient theory of single crystal plasticity. The theory is based on a system of microscopic force balances, one balance for each slip system, derived from the principle of virtual power, and a mechanical version of the second law that includes, via the microscopic forces, work performed during plastic flow. When combined with thermodynamically consistent constitutive relations the microscopic force balances become flow rules for the individual slip systems. Because these flow rules are in the form of partial differential equations requiring boundary conditions, they are nonlocal. The chief new ingredient in the theory is a free energy dependent on (geometrically necessary) edge and screw dislocation-densities as introduced in Gurtin [Gurtin, 2006. The Burgers vector and the flow of screw and edge dislocations in finite-deformation plasticity. Journal of Mechanics and Physics of Solids 54, 1882].     

An analysis of the pile-up of infinite periodic walls of edge dislocations

We analyse the equilibrium pile-up configurations of infinite periodic walls of edge dislocations which are forced against an impenetrable obstacle by a constant applied shear stress. Numerically generated density distributions exhibit two distinct regions, for each of which we provide an interpretation and an analytical prediction. Near the obstacle, the influence of neighbouring slip planes may be neglected and the classical solution for a single slip plane applies. At a larger distance a linear decay is obtained. The characteristic length scales of the two parts of the pile-up are shown to depend differently on the parameters of the problem.     

A screw dislocation by nonlinear continuum mechanics

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Residual stresses in thin film systems: Effects of lattice mismatch,thermal mismatch and interface dislocations

This paper explores the mechanisms of the residual stress generation in thin film systems with large lattice mismatch strain, aiming to underpin the key mechanism for the observed variation of residual stress with the film thickness. Thermal mismatch, lattice mismatch and interface misfit dislocations caused by the disparity of the material layers were investigated in detail. The study revealed that the thickness-dependence of the residual stresses found in experiments cannot be elucidated by thermal mismatch, lattice mismatch, or their coupled effect. Instead, the interface misfit dislocations play the key role, leading to the variation of residual stresses in the films of thickness ranging from 100 nm to 500 nm. The agreement between the theoretical analysis and experimental results indicates that the effect of misfit dislocation is far-reaching and that the elastic analysis of dislocation, resolved by the finite element method, is sensible in predicting the residual stress distribution. It was quantitatively confirmed that dislocation density has a significant effect on the overall film stresses, but dislocation distribution has a negligible influence. Since the lattice mismatch strain varies with temperature, it was finally confirmed that the critical dislocation density that leads to the measured residual stress variation with film thickness should be determined from the lattice mismatch strain at the deposition temperature.     

A continuum theory of structural change

Summary A comprehensive treatment of structural change (excluding the effects of induced anisotropy) is given which admits changes, and the associated variation of mechanical properties, from a variety of different influences. The treatment can accommodate the simultaneous effect of various influences and gives wider meaning and scope to the commonly used term thixotropy. A specific example of thixo-viscoelasticity is described in some detail. Most non dilute polymeric and solid-liquid systems can be expected to show some degree of thixotropy.     

A continuum theory for fibre-reinforced composites

The effective stiffness theory for fibre reinforced materials with a hexagonal layout of fibres is presented. The theory is illustrated by the dispersion curves of plane steadystate time-harmonic waves. The limiting phase velocities at vanishing wave numbers serve in the determination of the elastic moduli of the equivalent homogeneous transversely isotropic medium.     

A simple model for diffusion-induced dislocations during the lithiation of crystalline materials

Assuming that the lithiation reaction occurs randomly in individual small particles in the vicinity of the reaction front, a simple model of diffusion-induced dislocations was developed. The diffusion-induced dislocations are controlled by the misfit strain created by the diffusion of solute atoms or the phase transformation in the vicinity of the reaction front. The dislocation density is proportional to the total surface area of the “lithiated particle” and inversely proportional to the particle volume. The diffusion-induced dislocations relieve the diffusion-induced stresses.     

A continuum theory of time-dependent inelastic flow

Summary The simplest form of integral equations for describing time-dependent inelastic flow are derived. The nonlinear stress-strain rate relations introduce harmonics into the stress wave in oscillatory motion and if either the viscosity or yield criterion are time-dependent the stress harmonics will, in general, have a frequency dependent phase-angle relative to the strain rate. There is therefore the possibility of finite rotational or translational diffusion coefficients (which are not mentioned explicitly in the text) producing pseudo elastic effects in the non-linear region of response to oscillatory motion, even though there may be no deformable particles present.The rheological classes termed pseudoplastic, dilatant, thixotropic and rheopectic (antithixotropic) all fall within the scope of the treatment, depending upon the nature of the relaxation spectrum.     

A continuum theory of elastic material surfaces

A mathematical framework is developed to study the mechanical behavior of material surfaces. The tensorial nature of surface stress is established using the force and moment balance laws. Bodies whose boundaries are material surfaces are discussed and the relation between surface and body stress examined. Elastic surfaces are defined and a linear theory with non-vanishing residual stress derived. The free-surface problem is posed within the linear theory and uniqueness of solution demonstrated. Predictions of the linear theory are noted and compared with the corresponding classical results. A note on frame-indifference and symmetry for material surfaces is appended.     

Charged dislocations in piezoelectric bimaterials

In some piezoelectric semiconductors and ceramic materials, dislocations can be electrically active and could be even highly charged. However, the impact of dislocation charges on the strain and electric fields in piezoelectric and layered structures has not been presently understood. Thus, in this paper, we develop, for the first time, a charged three-dimensional dislocation loop model in an anisotropic piezoelectric bimaterial space to study the physical and mechanical characteristics which are essential to the design of novel layered structures. We first develop the analytical model based on which a line-integral solution can be derived for the coupled elastic and electric fields induced by an arbitrarily shaped and charged three-dimensional dislocation loop. As numerical examples, we apply our solutions to the typical piezoelectric AlGaN/GaN bimaterial to analyze the fields induced by charged square and elliptic dislocation loops. Our numerical results show that, except for the induced elastic (mechanical) displacement, charges along the dislocation loop could substantially perturb other induced fields. In other words, charges on the dislocation loop could significantly affect the traditional dislocation-induced stress/strain, electric displacement, and polarization fields in piezoelectric bimaterials.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

Trapping and escape of dislocations in micro-crystals with external and internal barriers

We perform three-dimensional dislocation dynamics simulations of solid and annular pillars, having both free-surface boundary conditions, or strong barriers at the outer and/or inner surfaces. Both pillar geometries are observed to exhibit a size effect where smaller pillars are stronger. The scaling observed is consistent with the weakest-link activation mechanism and depends on the solid pillar diameter, or the annular pillar effective diameter, D_eff = D − D_i, where D and D_i are the external and internal diameters of the pillar, respectively. An external strong barrier is observed to dramatically increase the dislocation density by an order of magnitude due to trapping dislocations at the surface. In addition, a considerable increase in the flow strength, by up to 60%, is observed compared to simulations having free-surface boundary conditions. As the applied load increases, weak spots form on the surface of the pillar by dislocations breaking through the surface when the RSS is greater than the barrier strength. The hardening rate is also observed to increase with increasing barrier strength. With cross-slip, we observe dislocations moving to other glide planes, and sometimes double-cross-slipping, producing a thickening of the slip traces at the surface. Finally the results are in qualitative agreement with recent compression experimental results of coated and centrally-filled micropillars.     

A two-dimensional continuum theory for angle-ply laminates

A two-dimensional continuum theory of microstructure is developed for stress analysis of angle-ply laminates under in-plane loading. An example problem is used to evaluate the results of the theory against a reference solution obtained by the finite element method. The results are in satisfactory agreement; they also show that the in-plane stresses reach somewhat higher peak values than reported in previous literature.The theory is also presented in a simplified version, which is found to be adequate for predicting interlaminar stresses and in-plane stress resultants, but does not give acceptable results for the variation of in-plane stresses through the thickness of the laminations.     

A continuum theory for viscoelastic composite beams

Summary A dynamical continuum theory is developed for laminated composite beams. Starting with an assumed displacement- and temperature field, the one-dimensional approximate theory is consistently constructed within the frame of the three-dimensional theory of linear, nonisothermal, anisotropic, coupled viscoelasticity. Each constituent of the beam may possess different constant thickness and mechanical properties. All dynamic interactions between the adjacent constituents are included. Further, the effects of transverse shear and normal strains and rotatory inertia as well as those of cross-sectional distortion are all taken into account. The resulting equations consist of the macroscopic beam equations of motion and heat conduction, the kinematical relations, the initial and boundary conditions and the constitutive equations, and they govern the extensional, flexural and torsional motions of laminated composite beams. The special cases of constituents which made of either isotropic thermoviscoelastic or anisotropic thermoelastic materials are discussed briefly.Supported by the Office of Naval Research.With 1 figure     

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.