Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part II

In Part I of this set of two papers, a model of mesoscopic plasticity is developed for studying initial-boundary value problems of small scale plasticity. Here we make qualitative, finite element method-based computational predictions of the theory. We demonstrate size effects and the development of strong inhomogeneity in simple shearing of plastically constrained grains. Non-locality in elastic straining leading to a strong Bauschinger effect is analyzed. Low shear strain boundary layers in constrained simple shearing of infinite layers of polycrystalline materials are not predicted by the model, and we justify the result based on an examination of the no-dislocation-flow boundary condition. The time-dependent, spatially homogeneous, simple shearing solution of PMFDM is studied numerically. The computational results and an analysis of continuous dependence with respect to initial data of solutions for a model linear problem point to the need for a nonlinear study of a stability transition of the homogeneous solution with decreasing grain size and increasing applied deformation. The continuous-dependence analysis also points to a possible mechanism for the development of spatial inhomogeneity in the initial stages of deformation in lower-order gradient plasticity theory. Results from thermal cycling of small scale beams/films with different degrees of constraint to plastic flow are presented showing size effects and reciprocal-film-thickness scaling of dislocation density boundary layer width. Qualitative similarities with results from discrete dislocation analyses are noted where possible.We discuss the convergence of approximate solutions with mesh refinement and its implications for the prediction of dislocation microstructure development, motivated by the notion of measure-valued solutions to conservation laws.     

Gradient single-crystal plasticity with free energy dependent on dislocation densities

This study develops a small-deformation theory of strain-gradient plasticity for single crystals. The theory is based on: (i) a kinematical notion of a continuous distribution of edge and screw dislocations; (ii) a system of microscopic stresses consistent with a system of microscopic force balances, one balance for each slip system; (iii) a mechanical version of the second law that includes, via the microscopic stresses, work performed during viscoplastic flow; and (iv) a constitutive theory that allows:

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the free energy to depend on densities of edge and screw dislocations and hence on gradients of (plastic) slip;

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the microscopic stresses to depend on slip-rate gradients.

The microscopic force balances when augmented by constitutive relations for the microscopic stresses results in a system of nonlocal flow rules in the form of second-order partial differential equations for the slips. When the free energy depends on the dislocation densities the microscopic stresses are partially energetic, and this, in turn, leads to backstresses in the flow rules; on the other hand, a dependence of these stresses on slip-rate gradients leads to a strengthening. The flow rules, being nonlocal, require microscopic boundary conditions; as an aid to numerical solutions a weak (virtual power) formulation of the flow rule is derived.     

Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div-curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.     

A non-singular continuum theory of dislocations

We develop a non-singular, self-consistent framework for computing the stress field and the total elastic energy of a general dislocation microstructure. The expressions are self-consistent in that the driving force defined as the negative derivative of the total energy with respect to the dislocation position, is equal to the force produced by stress, through the Peach-Koehler formula. The singularity intrinsic to the classical continuum theory is removed here by spreading the Burgers vector isotropically about every point on the dislocation line using a spreading function characterized by a single parameter a, the spreading radius. A particular form of the spreading function chosen here leads to simple analytic formulations for stress produced by straight dislocation segments, segment self and interaction energies, and forces on the segments. For any value a>0, the total energy and the stress remain finite everywhere, including on the dislocation lines themselves. Furthermore, the well-known singular expressions are recovered for a=0. The value of the spreading radius a can be selected for numerical convenience, to reduce the stiffness of the dislocation equations of motion. Alternatively, a can be chosen to match the atomistic and continuum energies of dislocation configurations.     

Analysis of dislocation nucleation from a crystal surface based on the Peierls-Nabarro dislocation model

Dislocation nucleation from a stressed crystal surface is analyzed based on the Peierls-Nabarro dislocation model. The variational boundary integral approach is used to obtain the profiles of the embryonic dislocations in various three-dimensional nucleation configurations. The stress-dependent activation energies required to activate dislocations from their stable to unstable saddle point configurations are determined. Compared to previous analyses of this type of problem based on continuum elastic dislocation theory, the present analysis eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Moreover, atomic information can be incorporated to reveal the dependence of the nucleation process on the profile of the atomic interlayer potential as compared to continuum elastic dislocation theory in which only elastic constants and Burgers vector are relevant. Finally, the presented methodology can also be readily used to study dislocation nucleation from the surface heterogeneities such as cracks, steps, and quantum structures of electronic devices.     

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ₂/μ₁.     

Correlation between structural and optical properties of ion beam synthesized β-FeSi2 precipitates in Si

The structural and optical properties of β-FeSi₂ precipitates produced by ion beam synthesis have been investigated by transmission electron microscopy, photoluminescence (PL) analysis and near infrared transmission measurements. The PL spectrum of β-FeSi₂ precipitates in a dislocation free sample has been observed to consist of a sharp line at 1.54 μm and a weak peak at 1.46 μm. Optical transmission measurements showed a direct band gap about 0.8 eV smaller than in continuous β-FeSi₂ film. Calculation of the electronic bands of β-FeSi₂ for different values of the lattice parameters indicates that this reduction can be ascribed to band distortion provided by the lattice strain.     

The effect of the location of stage-I fatigue crack across the persistent slip band on its growth rate – A 3D dislocation dynamics study

Abstract

A 3D dislocation dynamics study to ascertain the probable path of stage-I fatigue crack propagation across the persistent slip band (PSB) in austenitic stainless steel is presented. Cyclic plasticity and the resulting crack tip slip displacement (CTSD) are evaluated for cracks of varying length introduced at PSB-center and at two PSB-matrix interfaces. CTSD attains high value at either of the two interfaces irrespective of the proximity of crack front to the grain boundary. Further, a difference in microcrack propagation rate is also observed among the two interfaces. The present results assert microcrack propagation preferrentially along one of the two PSB-matrix interfaces rather than at the PSB-center. A pre-existing PSB dislocation structure localises the cyclic slip for crack lengths up to approximately half of the grain depth for an applied strain range of 2 × 10^?4.     

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO₃ ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.     

Microstructure evolution and mechanical property of pulsed laser welded Ni-based superalloy

For evaluating the microstructure evolution and mechanical property of Ni-based Hastelloy C-276 weld joint by the pulsed laser welding, the influence of pulsed laser welding on the microstructure and mechanical property of the weld joint is investigated by the analysis of the microstructure morphology, microhardness, phase structure and tensile property. The results indicate that, in the fusion zone three sections are divided on the basis of the patterns of grain structures. In the weld joint, the element segregation is found, but the trend of brittle phase׳s formation is weakened. The weld microhardness presents just a little higher than that of base metal, and there is no obvious the softened heat affected zone. Meanwhile in the weld joint, the phase structure is still the face-center cubic with the tiny shift of peak positions and widened Full Width at Half-Maximum. The yield strength of weld joint is the same as that of base metal, and the tensile strength is nearly 90% of that of base metal. The decreased tensile strength is mainly attributed to the dislocation piling-up.