Concentrated Point Force Research Articles

Continuum strain of point defects

We propose a new Regularized Green's Function Method (RGFM) derived from electron densities and captures the disturbance due to point defects, successfully extending the elastic strain determination to the lattice scales. The RGFM circumvents the use of concentrated point forces, which results in unrealistic singular fields. The objective is to determine the force variations at the atomic scales from Ab-initio calculations as Quantum Mechanical Force Density (QMFD). QMFD encodes the information regarding the electronic structure of the defect since it is related to the gradients of electron wavefunctions generated from the Density Functional Theory (DFT) calculations. Once the QMFD is calculated, RGFM solves the force equilibrium equation via Fourier transforms to compute elastic fields. Therefore, the present treatment reflects the electronic structure of the atomic positions, hence the complex elastic deformations and interactions at short range, which represents a significant advancement compared to previous studies. The RGFM can also capture the long-range fields since it calculates the decay of the force fields away from the nuclei. Previous theories, such as the elastic dipole method, face two main shortcomings: they can only handle the long-range elastic fields, and contain an unphysical singularity at the center of the point defect. Our novel derivation of distributed forces addresses both challenges, i.e., it renders non-singular displacement and strain fields at the nuclei, and can also describe the elastic response accurately far from the nuclei center. The application of the method to the NV (nitrogen-vacancy) center in SiC is demonstrated in comparison with the elastic dipole theory, showing the advantages of the present methodology.

On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing

The present two-part paper concerns the active vibration suppression for the simplest damped continuous system, namely the transverse oscillations of an elastic string, with constant tension and mass density per unit length and friction force proportional to the velocity, described by the telegraph or wave-diffusion equation, in two complementary parts. The initial part I considers non-resonant and resonant forcing, by concentrated point forces or continuous force distributions independent of time, with phase shift between the forced and free oscillations, in the absence of damping, in which case the forced telegraph equation reduces to the forced classical wave equation. The present and final part II uses the forced wave-diffusion equation to model the effect of damping, both as amplitude decay and phase shift in time, for non-resonant and resonant forcing by a single point force, with constant magnitude or magnitude decaying exponentially in time at an arbitrary rate. Assuming a finite elastic string fixed at both ends, the free oscillations are (i) sinusoidal modes in space-time with exponential decay in time due to damping. The non-resonant forced oscillations at an applied frequency distinct from a natural frequency are also (ii) sinusoidal in space-time, with constant amplitude and a phase shift such that the work of the applied force balances the dissipation. For resonant forcing at an applied frequency equal to a natural frequency, the sinusoidal oscillations in space-time have (iii) a constant amplitude and a phase shift of π/2. In both cases, the (ii) non-resonant or (iii) resonant forcing dominates the decaying free oscillations after some time. Even by optimizing the forcing to minimize the total energy of oscillation, it remains below the energy of the free oscillation alone, but only for a short time—generally a fraction of the period. A more effective method of countering the damped free oscillations is to use forcing with amplitude decaying exponentially in time; by suitable choice of the forcing decay relative to the free damping, the total energy of oscillation over all time can be reduced to no more than 1/16th of the energy of the free oscillation.

Free and forced in-plane vibration of rectangular plates with non-uniform elastic boundary conditions

In this paper, the free and forced in-plane vibration analysis of rectangular plates are performed for the first time using an improved Fourier series method, in which the boundary restraining spring stiffness can vary in any functional pattern along each edge. Two-dimensional improved Fourier series displacement forms are constructed with four supplementary polynomials introduced into the standard 2-D Fourier series to make the field functions sufficiently smooth in the whole solving domain. Energy formulations are employed to describe the in-plane dynamics of plate system, in which the in-plane concentrated point force is taken into account in the form of work term. All the unknown Fourier series coefficients are then solved through the Rayleigh-Ritz procedure. Several numerical examples are given to demonstrate the correctness and effectiveness of the proposed model through the comparison with those calculated via finite element analysis (FEA). The results show that these two results can agree very well with each other for various non-uniform boundary conditions. Based on the established model, the in-plane vibration response is also studied. Some curves and contours are obtained to illustrate how the boundary restraining stiffnesses affect the in-plane point and transfer mobility of rectangular plate structure.

Paradoxical properties of nonstationary wavefields of point sources in isotropic elastic medium

Components of the nonstationary Green tensor for a homogeneous, isotropic, elastic medium, which are solutions of concentrated point force problems, are considered. As is known, they may include anomalous wavefield, which is nonzero between the wave front sets of the P and S waves and grows, in general, with time. In the paper, the time dependencies of the source for which the anomalous wavefield does not arise, are characterized.

Free-Space Fundamental Solution of a 2D Steady Slow Viscous MHD Flow

The fundamental free-space 2D steady creeping MHD flow produced by a concentrated point force of strength g located at a so-called source point x0 in an unbounded conducting Newtonian liquid with uniform viscosity μ and conductivity σ>0 subject to a prescribed uniform ambient magnetic field B=Be1 is analytically obtained. More precisely, not only the produced flow pressure p and velocity u but also the resulting stress tensor field σ are expressed at any observation point x≠x0 in terms of usual modified Bessel functions, the vectors g,x−x0 and the so-called Hartmann layer thickness d=(μ/σ−−−√)/B (see Hartmann (1937)). The resulting basic flows obtained for g either parallel with or normal to the magnetic field B are examined and found to exhibit quite different properties.

A modified variational approach for vibration analysis of ring-stiffened conical–cylindrical shell combinations

Abstract This work presents a modified variational method for dynamic analysis of ring-stiffened conical–cylindrical shells subjected to different boundary conditions. The method involves partitioning of the stiffened shell into appropriate shell segments in order to accommodate the computing requirement of high-order vibration modes and responses. All essential continuity constraints on segment interfaces are imposed by means of a modified variational principle and least-squares weighted residual method. Reissner-Naghdi's thin shell theory combined with the discrete element stiffener theory to consider the ring-stiffening effect is employed to formulate the theoretical model. Double mixed series, i.e., the Fourier series and Chebyshev orthogonal polynomials, are adopted as admissible displacement functions for each shell segment. To test the convergence, efficiency and accuracy of the present method, both free and forced vibrations of non-stiffened and stiffened shells are examined under different combinations of edge support conditions. Two types of external excitation forces are considered for the forced vibration analysis, i.e., the axisymmetric line force and concentrated point force. The numerical results obtained from the present method show good agreement with previously published results and those from the finite element program ANSYS. Effects of structural damping on the harmonic vibration responses of the stiffened conical–cylindrical–conical shell are also presented.

Enhanced Sensitivity of Microcantilever Surface Stress Sensor

In this paper a novel L-shape microcantilever that is more sensitive in compare of traditional rectangular microcantilever is proposed. In traditional microcantilever surface stress sensor a slight momentum is induced by distributed area stress lead to a slight deformation. In contrast, the proposed sensor substitution the slight momentum with concentrated point force in lateral direction on the tip of the microcantilever. This lead to, mechanically amplifies of the influence of area stress and more displacement. In addition, it has been theoretically proved that proposed microcantilever sensor with same lateral area in compare with traditional microcantilever surface stress sensor can be 2 to 100 orders more sensitive. The proposed L-shape microcantilever configuration of sensor could be incorporated in most of the microfabrication processes. However in this paper we use a low cost fabrication method based on polymeric microcantilever with metal piezoresistive detection.

Finite elements modelling of scattering problems for flexural waves in thin plates: Application to elliptic invisibility cloaks, rotators and the mirage effect

We propose a finite elements algorithm to solve a fourth order partial differential equation governing the propagation of time-harmonic bending waves in thin elastic plates. Specially designed perfectly matched layers are implemented to deal with the infinite extent of the plates. These are deduced from a geometric transform in the biharmonic equation. To numerically illustrate the power of elastodynamic transformations, we analyze the elastic response of an elliptic invisibility cloak surrounding a clamped obstacle in the presence of a cylindrical excitation i.e. a concentrated point force. Elliptic cloaking for flexural waves involves a density and an orthotropic Young’s modulus which depend on the radial and azimuthal positions, as deduced from a coordinates transformation for circular cloaks in the spirit of Pendry et al. [Science 312, 1780 (2006)], but with a further stretch of a coordinate axis. We find that a wave radiated by a concentrated point force located a couple of wavelengths away from the cloak is almost unperturbed in magnitude and in phase. However, when the point force lies within the coating, it seems to radiate from a shifted location. Finally, we emphasize the versatility of transformation elastodynamics with the design of an elliptic cloak which rotates the wavevector of a flexural wave within its core.

Stress analysis in thin coatings and substrate subjected to point contact loading (mechanics of scratch test)

In this article, a model is developed to predict the stresses in thin coatings induced by a point concentrated load in continuation of articles in describing the tribological contact analysis of coated systems. This solution can be used as a suitable analytical model for mechanics of scratch test. Despite the lack of fully satisfactory analytical model of its mechanics, the scratch test is widely used to quantitatively evaluate the adhesion of coating and substrate. In this test an indenter is drawn across the surface of a coating under an increasing (continuous or step-wise) load. In order to develop an analytical model for scratch test in elastic zone, the fundamental solutions were obtained for concentrated point forces acting perpendicular and parallel to the layer surface. The stress and displacement fields were calculated for the coatings and the substrate due to these forces. These results were compared with the results of finite element method (FEM) by using ANSYS software, as well as with the results of uncoated elastic half-space. These results are also useful for ranking the coating–substrate adhesion of different coated systems, or for estimating the critical mean stress for interfacial failure.

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.

Elastodynamic response of a cracked orthotropic medium under impact loading

In this paper the problem of elastodynamic response of an infinite orthotropic medium containing a central crack under normal impact loading is considered. Laplace and Fourier integral transforms are employed to reduce the dimensional wave propagation problem to the solution of a pair of dual integral equations in the Laplace transform plane. These integral equations are then reduced to integro-differential equations which have been solved in the low frequency domain by method of iteration. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for normal point force loading. Numerical results of normalized SIF for large normalized time variable and for different concentrated point force loading at arbitrary location of the crack surface have been calculated for graphite–epoxy composite material. In the present paper the method of [A. Papoulis, The method of inversion of Laplace transform, Quart. Appl. Math. 14 (1957) 405–414] is used for the inversion of the Laplace transform. Finally the results obtained are displayed graphically.

Moving oscillating loads in 2D anisotropic elastic medium: Plane waves and fundamental solutions

The paper considers a concentrated point force moving with constant velocity and oscillating with constant frequency in an unbounded homogeneous anisotropic elastic 2D medium. Such problems come from the problems of a source that acts along the line in the corresponding three-dimensional anisotropic medium. Fundamental solutions for two-dimensional problems with moving oscillating sources lay the foundation for constructing solutions for more complicated problems and applying the boundary integral equation method. The properties of plane waves, their phase, slowness and ray or group velocity curves are determined in a moving coordinate system. The use of Fourier integral transform techniques and the properties of plane waves enables to obtain an explicit representation for elastodynamic Green’s tensor for all types of the source motion as a sum of integrals over a finite interval. The quasistatic and dynamic components of the Green’s tensor are obtained. The stationary phase method is employed to derive an asymptotic approximation of the far wave field. Simple formulae for the Poynting energy flux vectors for moving and stationary observers are also presented. It is noted that in far zones the wave fields are subdivided into separate cylindrical waves under kinematics and energy. It is shown that motion brings some difference in the far field properties, exemplified by the modification of the wave propagation zones and the change in their number, emergence of fast and slow waves under trans- and superseismic motion and etc.

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In present article we consider the problems of concentrated point force which is moving with constant velocity and oscillating with cyclic frequency in unbounded homogeneous anisotropic elastic two-dimensional medium. The properties of plane waves and their phase, slowness and ray or group velocity curves for 2D problem in moving coordinate system are described. By using the Fourier integral transform techniques and established the properties of the plane waves, the explicit representation of the elastodynamic Green's tensor is obtained for all types of source motion as a sum of the integrals over the finite interval. The dynamic components of the Green's tensor are extracted. The stationary phase method is applied to derive an asymptotic approximation of the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too. It is noted that in the far zones the cylindrical waves are separated under kinematics and energy. It is shown that the motion bring some differences in the far field properties. They are modification of the wave propagation zones and their number, fast and slow waves appearance under trans- and superseismic motion and so on.

The stress-intensity factor history for a half plane crack in a transversely isotropic solid due to impact point loading on the crack faces

Three-dimensional analysis is performed for a transversely isotropic solid containing a half plane crack subjected to suddenly applied concentrated point forces acting at a finite distance from the crack edge. The solution of this problem is treated as the superposition of two simpler problems. One considers the transient wave in an elastic half space generated by an impact point loading on the surface, the other problem is that which cancels out the surface displacement ahead of the crack edge induced by problem 1. A half space subjected to a distributed dislocation on the surface is constructed as the fundamental problem and solved by the use of integral transforms, the Wiener–Hopf technique and the Cagniard-de Hoop method. An exact expression is derived for the mode I stress-intensity factor as a function of time and position along the crack edge. Some features of the solution are discussed through numerical results.

A new insight into the reciprocity principle

Reciprocity is usually applied to wavefields associated with concentrated point forces and point receivers, but, reciprocity has a much wider application potential. In many cases, however, it is not used at its full potential because (1) a variety of source and receiver types are not considered or (2) its implementation is not well understood. We obtain reciprocity relations for inhomogeneous, anisotropic, viscoelastic solids and for distributed sources and receivers, and test these relations with a full‐wave numerical modeling algorithm. The theory and the numerical experiments show that, in addition to the usual relations involving directional forces, (1) the diagonal components of the strain tensor are reciprocal for dipole sources (single couple without moment), (2) the off‐diagonal components of the stress tensor are reciprocal for double couples with moments, (3) the dilatation due to a directional force is reciprocal to the particle velocity due to a dilatational source, and (4) some combinations of the off‐diagonal strains are reciprocal for single couples with moments.

Singular Solutions in Second-Order Elasticity

The problems of a concentrated point force in an infinite medium (Kelvin's problem), or applied normal to the boundary of a half-space (Boussinesq's problem), as well as the corresponding problems for a concentrated couple, are solved to second order for incompressible isotropic elastic solids. The solutions are based on a simplified form of the second-order equations, obtained previously, and use of the Love's function representation for axisymmetric problems in the linear theory.

Closed-form Green's functions for transversely isotropic bi-solids with a slipping interface

Green's functions are obtained in exact closed-forms for the elastic fields in bi-material elastic solids with slipping interface and differing transversely isotropic properties induced by concentrated point and ring force vectors. For the concentrated point force vector, the Green functions are expressed in terms of elementary harmonic functions. For the concentrated ring force vector, the Green functions are expressed in terms of the complete elliptic integral. Numerical results are presented to illustrate the effect of anisotropic bi-material properties on the transmission of normal contact stress and the discontinuity of lateral displacements at the slipping interface. The closed-form Green's functions are systematically presented in matrix forms which can be easily implemented in numerical schemes such as boundary element methods to solve elastic problems in computational mechanics.

Elastic fields in two joined transversely isotropic solids due to concentrated forces

This paper presents closed-form fundamental solutions for the elastic fields in two joined dissimilar transversely isotropic semi-infinite solids induced by concentrated forces. Two types of concentrated forces are considered in the paper. One is the forces concentrated at a point and the other is the forces uniformly distributed along a circle in the interior of the two joined dissimilar solids. The fundamental solutions are expressed in the forms of elementary harmonic functions and the complete elliptic integrals for the concentrated point forces and the concentrated ring forces, respectively. Numerical results show that the anisotropic bi-material properties have a significant effect on the singularities of the elastic fields when the forces are located at the bi-material interface.

Time domain dynamic response functions of elastically anisotropic solids

A method is described for calculating the displacement response function tensor for an infinite anisotropic elastic continuum subjected to a concentrated point force with Heaviside step function time dependence, and examples of computed response functions for zinc, diamond, and silicon are provided. Particular attention is given to the singularities of various orders that travel outward from the point of excitation on the wave surface. The commonly occurring forms that these wave arrival singularities take are described, and explanation provided of how they arise. Some of the singularity types mentioned here have not previously been noted in the literature. Computed response functions for silicon and zinc are compared with waveforms measured in these crystals using the capillary fracture technique.

Transient Analysis for Antiplane Crack Subjected to Dynamic Loadings

The problem considered here is the antiplane response of an elastic solid containing a half-plane crack subjected to suddenly applied concentrated point forces acting at a finite distance from the crack tip. A fundamental solution for the dynamic dislocation is obtained to construct the dynamic fracture problem containing a characteristic length. Attention is focused on the time-dependent full-field solutions of stresses and stress intensity factor. It is found that at the instant that the first shear wave reaches the crack tip, the stress intensity factor jumps from zero to the appropriate static value. The stresses will take on the appropriate static value instantaneously upon arrival of the shear wave diffracted from the crack tip, and this static value is thereafter maintained. The dynamic stress intensity factor of a kinked crack from this stationary semi-infinite crack after the arrival of shear wave is obtained in an explicit form as a function of the kinked crack velocity, the kink angle, and time. A perturbation method, using the kink angle as the perturbation parameter, is used. If the maximum energy release rate is accepted as the crack propagation criterion, then the crack will propagate straight ahead of the original crack when applying point load at the crack face.