Non-local Displacement Research Articles

Effective Optical Fiber Sensing Method for Stiffened Aircraft Structure in Nonlocal Displacement Framework

Effective Optical Fiber Sensing Method for Stiffened Aircraft Structure in Nonlocal Displacement Framework

Solutions for non-homogeneous wave equations subject to unusual and Neumann boundary conditions

This paper examines the accuracy, stability, and convergence of the forward and inverse solutions for non-homogeneous wave equations in different cases. The first case is when the wave governing equation is subject to the so-called unusual (non-local/non-classical) boundary condition. Where there is a non-local displacement and flux tension boundary data are placed at the very left end of the examined domain (string or rob), combined with Dirichlet and Neumann data at blue the left end of the domain. We explore the ideal value for the rational stiffness coefficient to be used for the second case which concerns moving Dirichlet data to the right and adding over-determination Neumann boundary conditions to the right or left. The forward solutions for the given problems, represented in the flux tension values, are investigated using the finite difference method (FDM) combined with the separation variable method. We also test the effect of the rotational stiffness coefficient on the effectiveness and convergence of those direct solutions. On the other hand, to obtain stable numerical results for the corresponding inverse problems (concerning retrieval of the force source function), the zeroth, the first, and the second-order of the Tikhonov regularization parameters are tested and the minimum error norm is calculated accordingly to find the best estimate for that space-dependent force source. Because such inverse problems are ill-posed, the existence and uniqueness of their solutions are tolerated and verified based on existing theorems in literature. This work includes numerical examples which illustrate the usefulness of the approach for solving non-homogeneous wave equations under novel kinds of boundary conditions.

A continuous‐discontinuous model for crack branching

SummaryA new continuous‐discontinuous model for fracture that accounts for crack branching in a natural manner is presented. It combines a gradient‐enhanced damage model based on nonlocal displacements to describe diffuse cracks and the extended finite element method (X‐FEM) for sharp cracks. Its most distinct feature is a global crack tracking strategy based on the geometrical notion of medial axis: the sharp crack propagates following the direction dictated by the medial axis of a damage isoline. This means that, if the damage field branches, the medial axis automatically detects this bifurcation, and a branching sharp crack is thus easily obtained. In contrast to other existing models, no special crack‐tip criteria are required to trigger branching. Complex crack patterns may also be described with this approach, since the X‐FEM enrichment of the displacement field can be recursively applied by adding one extra term at each branching event. The proposed approach is also equipped with a crack‐fluid pressure, a relevant feature in applications such as hydraulic fracturing or leakage‐related events. The capabilities of the model to handle propagation and branching of cracks are illustrated by means of different two‐dimensional numerical examples.

Stress-driven nonlocal integral elasticity for axisymmetric nano-plates

The stress-driven nonlocal integral model of elasticity for 1D nano-structures (Romano & Barretta, 2017a) is extended in this paper to Kirchhoff axisymmetric nano-plates. The nonlocal formulation, relating elastic principal flexural curvatures and moments, provides an effective methodology to assess size effects in 2D nano-structures. The associated elastostatic problem of nano-plates is conveniently expressed by differential relations equipped with constitutive boundary conditions involving nonlocal curvature fields. The proposed approach is illustrated by examining case-studies of engineering interest. In particular, nonlocal displacement solutions of axisymmetric nano-plates are detected for a variety of loading systems and kinematic boundary conditions. Merits and implications of the stress-driven strategy are elucidated by comparing the achieved results with those of the strain gradient model of elasticity generated by Reissner's variational principle. The outcomes can be useful for design and optimization of plate-like components of ground-breaking Nano-Electro-Mechanical-Systems (NEMS).

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C0 beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.

Elastic disturbance due to a nanoparticle near a free surface

The presence of nanoparticles in an elastic solid introduces disturbances that can vary significantly from the ones predicted by classical elasticity. In this study, we were able to determine that nanoscale effects introduced via a coherent interface model can indeed result in complex, non-local displacement and stress fields near the free surface of a substrate. The specific geometry is defined by a spherical nanoparticle near a straight boundary. The system is loaded either through a far-field uniaxial tension or a transformation strain (eigenstrain) in the particle itself. The elastic field can be fully determined using a three-dimensional displacement formulation that incorporates a well-established interface model.

Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions

In this paper, we study the linear bond-based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions. Both stationary and time-dependent problems are considered for a one-dimensional scalar equation defined on a finite bar and for a two-dimensional system defined on a square. The related peridynamic operators and associated functional spaces are analyzed. Applications to the numerical analysis of the finite-dimensional approximations to peridynamic models are also illustrated.

A new damage model based on non-local displacements

SUMMARY A new nonlocal damage model is presented. Nonlocality (of integral or gradient type) is incorporated into the model by means of nonlocal displacements. This contrasts with existing damage models, where a nonlocal strain or strain-related state variable is used. The new model is very attractive from a computational viewpoint, especially regarding the computation of the consistent tangent matrix needed to achieve quadratic convergence in Newton iterations. At the same time, its physical response is very similar to that of the standard models, including its regularization capabilities. All these aspects are discussed in detail and illustrated by means of numerical examples. Copyright c ∞ 2004 John Wiley & Sons, Ltd.

Non-local damage model based on displacement averaging

Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress–strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behaviour, various generalized continuum theories have been proposed. Integral-type non-local damage models are often based on weighted spatial averaging of a strain-like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the non-local displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analysed body, both formulations differ and the non-local displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non-local weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Monte Carlo simulation of freely jointed chains with variable bond lengths

AbstractMonte Carlo simulation of freely jointed off‐lattice chains with variable bond length is usually done with local random displacements of beads and with reptation moves (displacements of a bead along a chain). In dense systems, the acceptance ratio of reptations decreases strongly with density. We discuss versions of reptation moves, which are effective in dense systems. The idea, which comes from lattice systems, is to use a pseudovacancy (walker), which has the same size as a bead of a chain. The walker is attached to a neighbor chain and then another bead of that chain is cleaved. This is equivalent to a reptation move and a nonlocal displacement of the walker and since no free volume is needed, the move can be used with advantage in dense systems. A related technique are cooperative motions, which were introduced by T. Pakula for lattice models, where several chains change their conformation concomitantly. Such cooperative loops are implemented in the Monte Carlo algorithm by creating a temporary walker by cleaving a bead from a chain, moving it with reptations and finally annihilating the walker by attaching it to the same chain it was cleaved from. These moves and the condition of detailed balance are discussed in detail. As an example, we study the integrated autocorrelation time τint for the radius of gyration for a two‐dimensional system. For reduced densities larger than 0,4, we find that with standard reptations and local bead displacements τint increases strongly with density. If reptations with either a permanent or a temporary walker are used in addition to local moves, the integrated autocorrelation time changes only very little with density and very dense systems can still be simulated efficiently.