Journal of Micromechanics and Molecular Physics

The development of crystal plasticity theory based on dislocation patterns dynamics has been an outstanding problem in materials science and condensed matter of physics. Dislocation is the origin of crystal plasticity, and it is both the individual dislocation behavior as well as the aggregated dislocations behaviors that govern the plastic flow. The interactions among dislocations are complex statistical and stochastic events, in which the spontaneous emergence of organized dislocation patterns formations is the most critical and intriguing events. Dislocation patterns consist of quasi-periodic dislocation-rich and dislocation poor regions, e.g. cells, veins, labyrinths, ladders structures, etc. during cyclic loadings. Dislocation patterns have prominent and decisive effects on work hardening and plastic strain localization, and thus these dislocation micro-structures are responsible to material properties at macroscale. This paper reviews the recent developments of experimental observation, physical modeling, and computer modeling on dislocation microstructure. In particular, we focus on examining the mechanism towards plastic deformation. The progress and limitations of different experiments and modeling approaches are discussed and compared. Finally, we share our perspectives on current issues and future challenges in both experimental, analytical modeling, and computational aspects of dislocation pattern dynamics.

Recent experimental observations revealed the inherent nature of strong intermittent and heterogeneous plastic deformation at the nano- to micrometer scale. We present here a review of quantitative measures of temporal and spatial material instabilities associated with small-scale plastic flow. Spatial correlation characterization methods are developed and used to obtain information on the width of shear bands resulting from spatial instabilities. The effects of atomic-scale barriers to dislocation motion and the influence of sample size on temporal and spatial plastic instabilities are discussed. A simplified branching model of dislocation source activation is extended to predict dislocation barrier effects on strain burst statistics, and the transition from power law scaling to an exponential-like distribution. The connection between temporal and spatial plastic instabilities is discussed, and the efforts of considering these effects in crystal plasticity theory are also highlighted.

To allow for “relativistic”-like core contraction effects, an anisotropic regularization of steadily moving straight dislocations of arbitrary orientation is introduced, with two scale parameters $a_{\parallel }$ and $a_{\perp }$ along the direction of motion and transverse to it, respectively. The dislocation core shape is an ellipse. When $a_{\perp }/a_{\parallel }\to 0$, the model reduces to the Peierls–Eshelby dislocation, the fields of which are non-differentiable on the slip plane. For finite $a_{\parallel }$ and $a_{\perp }$, fields are everywhere differentiable. Applying the author’s so-called “causal” Stroh formalism to the model, explicit expressions for the regularized fields in anisotropic elasticity are derived for any velocity. For faster-than-wave velocities, Mach-cone angles are found insensitive to the ratio $a_{\parallel }/a_{\perp }$, as must be. However, the larger $a_{\parallel }$, the weaker the intensity of the cone branches. An expression is given for the radiative dissipative force opposed to motion. From this expression, it is inferred that the concept of a “radiation-free” intersonic velocity can, when not applicable, be replaced by that of a “least-radiation” velocity.

Crowdion defects are produced in body-centered-cubic metals under irradiation. Their structure and diffusive dynamics play a governing role in microstructural evolution, and hence the mechanical properties of nuclear materials. In this paper, we apply the analytical Frenkel-Kontorova model to crowdions and clusters thereof (prismatic dislocation loops) and show that the Peierls potential in which these defects diffuse is remarkably small (in the micro eV range as compared to the eV range for other defects). We also develop a coarse-grained statistical methodology for simulating these fast-diffusing objects in the context of object kinetic Monte Carlo, which is less vulnerable to the low barrier problem than naïve stochastic simulation.

Spectral methods using Fast Fourier Transform (FFT) algorithms have recently seen a surge in interest in the mechanics community. The present contribution addresses the critical question of determining local mechanical fields using the FFT method in the presence of interfacial defects. Precisely, the present work introduces a numerical approach based on intrinsic discrete Fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of disclinations, i.e., rotational discontinuities, and inhomogeneities. A centered finite difference scheme for differential rules are first used for numerically solving the Poisson-type equations in the Fourier space to get the incompatible elastic fields due to disclinations and dislocations. Second, centered finite differences on a rotated grid are chosen for the computation of the modified Fourier-Green’s operator in the Lippmann–Schwinger–Dyson type equation for heterogeneous media. Elastic fields of disclination dipole distributions interacting with inhomogeneities of varying stiffnesses, grain boundaries seen as DSUM (Disclination Structural Unit Model), grain boundary disconnection defects and phase boundary “terraces” in anisotropic bi-materials are numerically computed as applications of the method.

This paper discusses a coupled mechanics–phase-field model that can predict microstructure evolution in metallic polycrystals and in particular evolution of lattice orientation due to either deformation or grain boundary migration. The modeling framework relies on the link between lattice curvature and geometrically necessary dislocations and connects a micropolar or Cosserat theory with an orientation phase-field model. Some focus is placed on the underlying theory and in particular the theory of dislocations within a continuum single crystal plasticity setting. The model is finally applied to the triple junction problem for which there is an analytic solution if the grain boundary energies are known. The attention is drawn on the evolution of skew–symmetric stresses inside the grain boundary during migration.

A constitutive framework based on concepts from phase field theory and pseudo-Finsler geometry is exercised in numerical simulations of deformation and fracture of ceramic polycrystals. The material system of interest is boron carbide, a hard but brittle ceramic. Some microstructures are enabled with thin layers of a secondary amorphous phase of boron nitride between grains of boron carbide. The constitutive theory accounts for physical mechanisms of twinning, crystal-to-glass phase transformations, cleavage fracture within grains and separation and cavitation at grain boundaries (GBs). According to the generalized Finsler approach, geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors, that enter the energy potential in a manner similar to order parameters of phase field models. A partially linearized version of the theory is invoked in finite element simulations of polycrystals, with and without GB layers, subjected to pure shear loading. Effects of grain size and layer properties — thickness, shear modulus and surface energy — are studied parametrically. Results demonstrate that twinning and amorphization occur prominently in nanocrystals but less so in aggregates with larger grains that tend to fail earlier by fracture. Structural changes occur readily in the latter at smaller applied strains only in conjunction with elastic shear softening in localized degraded or damaged regions. Hall–Petch scaling of peak shear strength with grain size is observed. Strength is increased via addition of amorphous layers that shift the failure mode from transgranular to intergranular and further by cavity expansion in layers that induces local elastic compression and suppresses crack extension. Stiff layers provide the largest peak strength enhancement, while elastically compliant layers may improve toughness and strength in the softening regime.

A new model for determining band gaps for elastic wave propagation in a periodic composite beam structure is developed using a non-classical Timoshenko beam model that incorporates the surface energy, transverse shear and rotational inertia effects. The Bloch theorem and transfer matrix method for periodic structures are employed in the formulation. The new model reduces to the classical elasticity-based model when the surface energy effect is not considered. It is shown that the band gaps predicted by the current model depend on the surface elastic constants of each constituent material, beam thickness, unit cell size, and volume fraction. The numerical results reveal that the band gap based on the current non-classical model is always larger than that given by the classical model when the beam thickness is very small, but the difference is diminishing as the thickness becomes large. Also, it is found that the first frequency for producing the band gap and the band gap size decrease with the increase of the unit cell length according to both the current and classical models. In addition, it is observed that the volume fraction has a significant effect on the band gap size, and large band gaps can be obtained by tailoring the volume fraction and material parameters.

The purpose of this article is to compare concepts of defect nucleation based on bifurcation of equilibrium solution and on damage modelling. The nucleation criterion is defined as a bifurcation of the equilibrium solutions of the perfect body and of the imperfect one when the size of the defect vanishes. The defect is considered as a small volume which evolves as a damaged zone. To study the influence of geometry of the defect on the critical loading governing its initiation, we consider the particular cases of a linear elastic composite sphere and of a linear elastic composite cylinder, for which the equilibrium solutions are known when the radial distribution of elastic bulk modulus is given simultaneously with a uniform shear modulus. The initial defect is a small sphere or a small cylinder, respectively, it can be a cavity or a kernel made with an elastic material with lower mechanical properties.

In this work, interaction problems between a finite-length crack with plane and antiplane crystal defects in the context of couple-stress elasticity are presented. Two alternative yet equivalent approaches for the formulation of crack problems are discussed based on the distributed dislocation technique. To this aim, the stress fields of climb and screw dislocation dipoles are derived within couple-stress theory and new ‘constrained’ rotational defects are introduced to satisfy the boundary conditions of the opening mode problem. Eventually, all interaction problems are described by single or systems of singular integral equations that are solved numerically using appropriate collocation techniques. The obtained results aim to highlight the deviation from classical elasticity solutions and underline the differences in interactions of cracks with single dislocations and dislocation dipoles. In general, it is concluded that the cracked body behaves in a more rigid way when couple-stresses are considered. Also, the stress level is significantly higher than the classical elasticity prediction. Moreover, the configurational forces acting on the defects are evaluated and their dependence on the characteristic material length of couple-stress theory and the distance between the defect and the crack-tip is discussed. This investigation reveals either a strengthening or a weakening effect in the opening mode problem while in the antiplane mode a strengthening effect is always obtained.

The theory of Mindlin’s isotropic strain gradient elasticity of form II is reviewed. Three-dimensional and two-dimensional Green tensors and their first and second derivatives are derived for an unbounded medium. Using an operator-split in Mindlin’s strain gradient elasticity, three-dimensional and two-dimensional regularization function tensors are computed, which are the three-dimensional and two-dimensional Green tensors of a tensorial Helmholtz equation. In addition, a length scale tensor is introduced, which is responsible for the characteristic material lengths of strain gradient elasticity. Moreover, based on the Green tensors of Mindlin’s strain gradient elasticity, point, line and double forces are studied.

An account of non-singular solutions of gradient elasticity (GradEla) models for dislocations, along with clarifications of certain issues raised on previously published results, is given. Then, an extension to fractional GradEla solutions is pursued and certain preliminary results on this emerging topic are listed.

In this work, we derive the $J$-, $M$- and $L$-integrals of body charges and point charges in electrostatics, and the $J$-, $M$- and $L$-integrals of body forces and point forces in elasticity and we investigate their physical interpretation. Electrostatics is considered as field theory of an electrostatic scalar potential $\phi$ (scalar field theory) and elasticity as field theory of a displacement vector $u$ (vector field theory). One of the basic quantities appearing in the $J$-, $M$- and $L$-integrals is the electrostatic Maxwell–Minkowski stress tensor in electrostatics and the Eshelby stress tensor in elasticity. Among others, it is shown that the $J$-integral of body charges in electrostatics represents the electrostatic part of the Lorentz force, and the $J$-integral of body forces in elasticity represents the Cherepanov force. The $M$-integral between two-point sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity between these two-point sources. The $L$-integral represents the configurational vector moment or torque between two body or point sources (charges or forces). Interesting mathematical and physical features are revealed through the connection of the $J$-, $M$- and $L$-integrals with their corresponding infinitesimal generators in both theories. Several important outcomes arise from the comparison between the examined concepts in electrostatics and elasticity. Differences and similarities, that provide a deeper insight into the $J$-, $M$- and $L$-integrals and the related quantities to them, are pointed out and discussed. The presented results show that the $J$-, $M$- and $L$-integrals are fundamental concepts which can be applied in any field theory.