In this article, we introduce a new homotopy function to trace the trajectory by applying modified homotopy continuation method for finding the solution of the Linear Complementarity Problem. Earlier several authors attempted to propose homotopy func
tions based on original problems. We propose the homotopy function based on the Karush-Kuhn-Tucker condition of the corresponding quadratic programming problem of the original problem. The proposed approach extends the processability of the larger class of linear complementarity problems and overcomes the limitations of other existing homotopy approaches. We show that the homotopy path approaching the solution is smooth and bounded. We find the positive tangent direction of the homotopy path. The difficulty of finding a strictly feasible initial point for the interior point algorithm can be replaced appropriately by combining the interior point with the homotopy method. Various classes of numerical examples are illustrated to show the effectiveness of the proposed algorithm.
We revisit the class of column competent matrices and study some matrix theoretic properties of this class. The local $w$-uniqueness of the solutions to the linear complementarity problem can be identified by the column competent matrices. We establi
sh some new results on $w$-uniqueness properties in connection with column competent matrices. These results are significant in the context of matrix theory as well as algorithms in operations research. We prove some results in connection with locally $w$-uniqueness property of column competent matrices. Finally we establish a connection between column competent matrices and column adequate matrices with the help of degree theory.
Pinning of dislocations at nanosized obstacles like precipitates, voids and bubbles, is a crucial mechanism in the context of phenomena like hardening and creep. The interaction between such an obstacle and a dislocation is often explored at fundamen
tal level by means of analytical tools, atomistic simulations and finite element methods. Nevertheless, the information extracted from such studies has not been utilized to its maximum extent on account of insufficient information about the underlying statistics of this process comprising a large number of dislocations and obstacles in a system. Here we propose a new statistical approach, where the statistics of pinning of dislocations by idealized spherical obstacles is explored by taking into account the generalized size-distribution of the obstacles along with the dislocation density within a three-dimensional framework. The application of this approach, in combination with the knowledge of fundamental dislocation-obstacle interactions, has successfully been demonstrated for dislocation pinning at nanovoids in neutron irradiated type 316-stainless steel in regard to both conservative and non-conservative motions of dislocations.
By means of atomistic simulations, we demonstrate that a dislocation core exhibits intermittent quasistatic restructuring during incremental shear within the same Peierls valley. This can be regarded as a stick-slip transition, which is also reproduc
ed for a one-dimensional Frenkel-Kontorova chain under rigid boundary conditions. This occurs due to a discontinuous jump in an order parameter of the system, which signifies the extent of region forbidden for the presence of particles in the chain. The stick-slip phenomenon observed in the dislocation core is also shown to be reflected after dimensionality reduction of the multidimensional atomic coordinates, which provides a basis for comparison with the simple one-dimensional chain.
We employ the methods of atomistic simulation to investigate the climb of edge dislocation at nanovoids by analyzing the energetics of the underlying mechanism. A novel simulation strategy has been demonstrated to estimate the release of surface ener
gy of the nanovoid during the void induced climb. The curvature of the pinned dislocation segment is found to play a key role in mediating this unique mechanism of dislocation climb. Our study reveals that the kinetics of void-induced climb process is fundamentally distinct from the conventional diffusion-mediated climb.
We carry out the molecular statics simulations of depinning of an edge dislocation at voids of diameters of the order of ~1 nm. We show that the dislocation-void interactions are non-trivial as the applied shear load is found to enhance the pinning s
trength of the nanovoid itself. This leads to the surprising observation of multiple CRSS values for a given nanovoid.
Dislocation pinning plays a vital role in the plastic behaviour of a crystalline solid. Here we report the first observation of the damped oscillations of a mobile dislocation after it gets pinned at an obstacle in the presence of a constant static s
hear load. These oscillations are found to be inertial, instead of forced as obtained in the studies of internal friction of solid. The rate of damping enables us to determine the effective mass of the dislocation. Nevertheless, the observed relation between the oscillation frequency and the link length is found to be anomalous, when compared with the theoretical results in the framework of Koehlers vibrating string model. We assign this anomaly to the improper boundary conditions employed in the treatment. Finally, we propose that the inertial oscillations may offer a plausible explanation of the electromagnetic emissions during material deformation and seismic activities.
Plastic deformation of crystals is a physical phenomenon, which has immensely driven the development of human civilisation since the onset of the Chalcolithic period. This process is primarily governed by the motion of line defects, called dislocatio
ns. Each dislocation traps a quantum of plastic deformation expressible in terms of its Burgers vector[1,2]. Theorising the mechanisms of dislocation motion at the atomistic scales of length and time remains a challenging task on account of the extreme complexities associated with the dynamics. We present a new concept of modelling a moving dislocation as the dynamic distribution of the elastic field singularity within the span of the Burgers vector. Surprisingly, numerical implementation of this model for the periodic expansion-shrinkage cycle of the singularity is found to exhibit an energetics, which resembles that of a dislocation moving in the presence of the Peierls barrier[1-4]. The singularity distribution is shown to be the natural consequence under the external shear stress. Moreover, in contrast to the conventional assumption, here the calculations reveal a significant contribution of the linear elastic region surrounding the core towards the potential barrier.
This paper has been withdrawn.
In this letter we propose a model that demonstrates the effect of free surface on the lattice resistance experienced by a moving dislocation in nanodimensional systems. This effect manifests in an enhanced velocity of dislocation due to the proximity
of the dislocation line to the surface. To verify this finding, molecular dynamics simulations for an edge dislocation in bcc molybdenum are performed and the results are found to be in agreement with the numerical implementations of this model. The reduction in this effect at higher stresses and temperatures, as revealed by the simulations, confirms the role of lattice resistance behind the observed change in the dislocation velocity.
