Time-harmonic Loading Research Articles

Elastodynamic analysis of anisotropic elastic solid with multiple nanocavities

Abstract2D elastodynamic problem for a finite elastic anisotropic solid with nanosized cavities is worked out in this paper. The solid under plane strain conditions is subjected to time-harmonic load. The developed computational tool for the steady-state problem is boundary integral equation method based on frequency-dependent fundamental solutions for elastic anisotropic continuum obtained by Radon transform. Accuracy of the dynamic stress concentration factor and displacement components is proven by comparisons with existing in the literature analytical solution and with the results obtained using commercial finite element package. In addition, a parametric study is performed in order to examine the sensitivity of the dynamic stress concentration field to the geometrical configuration of multiple cavities, to the characteristics of the surface elastic properties, to the type of orthotropy and to the dynamic interaction between cavities.

Elastic solutions due to a time-harmonic point load in isotropic multi-layered media

A new analytical derivation of the elastodynamic point load solutions for an isotropic multi-layered half-space is presented by means of the precise integration method (PIM) and the approach of dual vector. The time-harmonic external load is prescribed either on the external boundary or in the interior of the solid medium. Starting with the axisymmetric governing motion equations in a cylindrical coordinate system, a second order ordinary differential matrix equation can be gained by making use of the Hankel integral transform. Employing the technique of dual vector, the second order ordinary differential matrix equation can be simplified into a first-order one. The approach of PIM is implemented to obtain the solutions of the ordinary differential matrix equation in the Hankel integral transform domain. The PIM is a highly accurate algorithm to solve sets of first-order ordinary differential equations and any desired accuracy of the dynamic point load solutions can be achieved. The numerical simulation is based on algebraic matrix operation. As a result, the computational effort is reduced to a great extent and the computation is unconditionally stable. Selected numerical trials are given to validate the accuracy and applicability of the proposed approach. More examples are discussed to portray the dependence of the load-displacement response on the isotropic parameters of the multi-layered media, the depth of external load and the frequency of excitation.

Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions

In this paper, a non-iterative technique is proposed for the transverse load identification on Kirchhoff plates using approximate Green’s functions (AGFs). In this way, we firstly employ the recently introduced meshless method to construct the AGFs, as the combination of a series of Trefftz basis, i.e. Exponential basis functions (EBFs), and the fundamental solutions of the governing equation. As will be explained, using a proper set of EBFs, as well as a collocation technique, enables us to construct the AGFs for different types of domain shape and boundary conditions. In the second step, a set of artificially generated results, in the absence of realistic experimental results, are used to express the plate’s response field, i.e. deflection or velocity fields, as a series of AGFs through a collocation technique. It will be shown how the constant coefficients of the response series are related to the intensity of the reconstructed force at a set of selected points. The proposed method is capable of constructing both distributed and concentrated loads with desirable accuracy. This ability is shown in the solution of three sample problems of the static and time-harmonic force recovery.

Dynamic analysis of a vertically loaded rigid disc in a transversely isotropic multilayered half-space

This paper focuses on the investigation of the response of a vertically loaded rigid circular disc embedded in a transversely isotropic multilayered half-space. The solutions for this dynamic interaction problem are obtained by utilizing the boundary element method, which takes the analytical layer-element solution of the transversely isotropic multilayered half-space subjected to an interior ring time-harmonic loading as the fundamental solution. The accuracy of present solutions is verified by the comparisons with existing solutions. In addition, selected numerical results are presented to illustrate the influence of embedded depth, frequency of excitation, material anisotropy and layering of the embedding medium on vertical impedance and resultant contact stresses.

Solutions for Triangularly Distributed Harmonic Loadings on Axially Symmetric Area in Layered Half-Space

ABSTRACTThe paper is to solve the problem of the response of stratified half-space subjected to triangularly distributed time-harmonic loadings on an axial symmetric area in the stratified half-space. Since the distributed area can be very small, the solution can be employed to deal with the problems of elastodynamics using the fundamental solution method. The advantage of the presented solution over Green function is that no singularity will occur in the presented solution. Some numerical results are given and some conclusions have been made. The presented solution is very efficient in computation.

In-plane stress analysis of cracked layers under harmonic loads

Stress analysis is carried out in an isotropic layer weakened by multiple cracks under time-harmonic loading. The viscous damping is used to model energy dissipation in the material. The analysis is based on the stress fields caused by climb and glide of an edge dislocation in the layer. The solution for dislocation is obtained by means of the integral transform method. Furthermore, stress analysis in the intact layer under self-equilibrating harmonic point loads is carried out. These solutions are employed to derive a set of Cauchy singular integral equations for analyzing cracks parallel/perpendicular to the layer boundary. The numerical solution of these equations yields dislocation densities on a crack surface which are used to determine dynamic stress intensity factors over a range of frequency. The natural frequencies of layers with horizontal cracks are obtained and the interaction of multiple cracks is studied.

Performance Analysis of Multi-GPU Implementations of Krylov-Subspace Methods Applied to FEA of Electromagnetic Phenomena

We present a comparison of performances of various graphic processing unit (GPU)-CUDA implementations of preconditioned Krylov-subspace methods, showing the impact of using a multi-GPU configuration. We aim to show that this resource allows the massively parallelized solution of large-scale real-world problems in state-of-the-art desktop PCs, since it overcomes the low-memory capacity of a single GPU, while still providing significant speedups when compared with either the usual sequential execution on a single-core CPU or an OpenMP implementation with four cores. The methods were selected based on their suitability to solve large-scale systems of equations arising from the 3-D finite-element analysis of open-bound earth current diffusion problems, both in steady state and under time-harmonic loading. In the latter, an ungauged harmonic edge-element formulation using the magnetic vector potential and the electric scalar potential was used. As the preconditioners suitable to this case, based on incomplete factorizations, are not appropriate for parallelization, we propose a hybrid CPU-GPU scheme to solve such problems, which still exhibits a competitive performance in low-cost PC desktops.

Bandgaps and directional propagation of elastic waves in 2D square zigzag lattice structures

In this paper we propose various types of two-dimensional (2D) square zigzag lattice structures, and we study their bandgaps and directional propagation of elastic waves. The band structures and the transmission spectra of the systems are calculated by using the finite element method. The effects of the geometry parameters of the 2D-zigzag lattices on the bandgaps are investigated and discussed. The mechanism of the bandgap generation is analyzed by studying the vibration modes at the bandgap edges. Multiple wide complete bandgaps are found in a wide porosity range owing to the separation of the degeneracy by introducing bending arms. The bandgaps are sensitive to the geometry parameters of the systems. The deformed displacement fields of the transient response of finite structures subjected to time-harmonic loads are presented to show the directional wave propagation. The research in this paper is relevant to the practical design of cellular structures with enhanced vibro-acoustics performance.

Optimization of nonlinear structural resonance using the incremental harmonic balance method

We present an optimization procedure for tailoring the nonlinear structural resonant response with time-harmonic loads. A nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis. An optimization procedure based on a gradient-based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities. We consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super-harmonic resonant responses.

Interface behavior of a bi‐material plate under dynamic loading. Cohesive interface debonding

The paper deals with the elastic and cohesive interface behavior of pre‐cracked bi‐material ceramic‐metal structures under dynamic time harmonic load. The shear lag model as well as the Fourier method is applied to find the dynamic response of the considered bi‐material structure, assuming the cohesive interface behaviour, accompanied before of the elastic‐brittle one. In both cases, the growth of debond length is not considered, e.g. at a given loading condition the only corresponding debond length is found. The inertia forces of the already elastic debond parts of the bi‐material structure are neglected. Appropriate contact conditions are proposed in order to fit together both elastic and cohesive solutions. The numerical predictions for the cohesive debond length of the bi‐material structures is calculated by the aid of the corresponding value of the elastic debond length at the same loading condition. The influence of loading characteristics i.e. frequencies and amplitude fluctuations on the debond length and the interface shear stress distribution is discussed. The parametric analysis of the results obtained is illustrated by examples of the modern ceramic‐metal composites on metal substrates and is depicted in figures.

Dynamic analysis of a rigid circular foundation on a transversely isotropic half-space under a buried inclined time-harmonic load

The dynamic analysis of a surface rigid foundation in smooth contact with a transversely isotropic half-space under a buried inclined time-harmonic load is addressed. By virtue of the superposition technique, appropriate Green׳s functions, and employing further mathematical techniques, solution of the mixed-boundary-value problem is expressed in terms of two well-known Fredholm integral equations. Two limiting cases of the problem corresponding to the static loading and isotropic medium are considered and the available results in the literature are fully recovered. For the static case, the results pertinent to both frictionless and bonded contacts are obtained and compared. With the aid of the residue theorem and asymptotic decomposition method, an effective and robust approach is proposed for the numerical evaluation of the obtained semi-infinite integrals. For a wide range of the excitation frequency, both normal and rotational compliances are depicted in dimensionless plots for different transversely isotropic materials. Based on the obtained results, the effects of anisotropy are highlighted and discussed.

Dynamic Torsional Response of a Pile Partially Embedded in Saturated Soil

The dynamic response of an elastic supporting pile partially embedded in a saturated soil and subjected to a time-harmonic torsional loading is investigated. At first, the pile is divided into two parts along the vertical direction, pile part above the soil and pile part embedded in the soil. Then, based on boundary and continuity conditions of the pile-soil system, the torsional impedance at the top end of the pile part embedded in the soil is obtained. By utilizing the transfer technique of impedance function, the admittance function of the pile top is defined in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, a semi-analytical solution for the velocity response of a pile subjected to a semi-sine wave exciting torque is obtained in the time domain. Finally, selected numerical results are obtained to analyze the influence of main parameters on the torsional vibration characteristics of the pile.

Dynamic behavior of several cracks in functionally graded strip subjected to anti-plane time-harmonic concentrated loads

This paper contains a theoretical formulations and solutions of multiple cracks subjected to an anti-plane time-harmonic point load in a functionally graded strip. The distributed dislocation technique is used to construct integral equations for a functionally graded material strip weakened by several cracks under anti-plane time-harmonic load. These equations are of Cauchy singular type at the location of dislocation, which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to evaluate the stress intensity factor and strain energy density factors (SEDFs) for multiple cracks with different configurations. Numerical calculations are presented to show the effects of material properties and the crack configuration on the dynamic stress intensity factors and SEDFs of the functionally graded strip with multiple curved cracks.

Analysis of Forced Vibrations in Micro-Scale Anisotropic Thermo-Elastic Beams Due to Concentrated Loads

This article is devoted to model and analyze the transverse deflection and thermal moment of transverse vibrations in a transversely isotropic, thermo-elastic beam resonator under the action of time harmonic concentrated load. The governing equations of flexural vibrations and thermal moment in a transversely isotropic, thermo-elastic Euler–Bernoulli beam have been derived in a closed form. A time harmonic point load is assumed to act on the beam at a given distance from the origin. The beam is assumed to be at either clamped-clamped (CC), simply supported-simply supported (SS), clamped-simply supported (CS), or clamped-free (CF) conditions at its ends. The Laplace transform technique has been used to find the transverse deflection and thermal moment in the transform domain due to the action of concentrated load in a beam under above conditions in case of both free and forced vibrations. The analytic expressions obtained in the physical domain after inversion of Laplace transforms have been computed numerically with the help of MATLAB software for silicon material beam. The results for coupled thermo-elastic, elastic, and isotropic beams have been deduced as special cases. The computer-simulated results have been presented graphically. The study may find applications in development and design of resonators (sensors), precision thermometers, and energy harvesters.

Effects of geometric and material properties on electrical power harvested from a bimorph piezoelectric cantilever beam

Purpose – A three-dimensional finite element (FE) model is developed to design a vibrating bimorph piezoelectric cantilever beam with lead zirconate titanate (PZT-5H) for energy harvesting. The paper aims to discuss these issues. Design/methodology/approach – A parametric study of electric power generated as a function of the dielectric constant, transverse piezoelectric strain constant, length and thickness of the piezoelectric material, is conducted for a time-harmonic surface pressure load. Transversely isotropic elastic and piezoelectric properties are assigned to the bimorph layers with brass chosen as the substrate material in the three-dimensional FE model. Using design of experiments, a study was conducted to determine the sensitivity of power with respect to the geometric and material variables. Findings – The numerical analysis shows that a uniform decrease in thickness and length coverage of the piezoelectric layers results in a nonlinear reduction in power amplitude, which suggests optimal values. The piezoelectric strain coefficient, d31 and the thickness of PZT-5H, tp, are the most important design parameters to generate high electric energy for bimorph vibration harvesting device. Originality/value – The work demonstrates that, through a sensitivity analysis, the electro-mechanical piezoelectric coupling coefficient (d31) and the thickness of the piezoelectric strips (tp) are the most important parameters which have a significant effect on power harvested.

Dynamic Crack Analysis in Layered Piezoelectric Composites under Time-Harmonic Loading

In this Paper, Time-Harmonic Dynamic Crack Analysis in Two-Dimensional (2D), Layered and Linear Piezoelectric Composites is Presented. A Frequency-Domain Symmetric Galerkin Boundary Element Method (SGBEM) is Developed for this Purpose. the Piecewise Homogeneous Sub-Layers of the Piezoelectric Composites are Modeled by the Multi-Domain BEM Formulation. the Frequency-Domain Dynamic Fundamental Solutions for Linear Piezoelectric Materials are Applied in the Present BEM. the Boundary Integral Equations are Solved Numerically by a Galerkin-Method Using Quadratic Elements. an Iterative Solution Algorithm is Implemented to Consider the Non-Linear Semi-Permeable Electrical Crack-Face Boundary Conditions. Numerical Examples will be Presented and Discussed to Show the Influences of the Location and Size of the Crack, the Material Combination of the Sub-Layers, the Piezoelectric Effect and the Time-Harmonic Dynamic Loading on the Dynamic Intensity Factors.

Waves and energy in random elastic guided media through the stochastic wave finite element method

Energy propagation in random viscoelastic media is considered in this Letter. The forced response of uncertain waveguide subject to time harmonic loading is treated. This energy model is based on a spectral approach called the “Stochastic Wave Finite Element” (SWFE) method which is detailed in this Letter. Assuming that the random properties are spatially homogeneous in the media, the SWFE is a hybridization of the deterministic wave finite element and a parametric probabilistic approach. The proposed model is applicable in a wide frequency band with reduced time consumption. Numerical examples show the effectiveness of the proposed approach to predict the statistics of kinematic and quadratic variables of guided wave propagation. The results are compared to Monte Carlo simulations.

Dynamic fracture analysis of multiple defects in an imperfect FGM coating-substrate layers

Fracture behavior of an orthotropic strip with an orthotropic functionally graded coating weakened by multiple defects under time-harmonic excitation is investigated in this paper. Imperfect bonding between layers is assumed and the problem is divided to analytical and computational parts using distributed dislocation technique. The dislocation densities are employed to determine dynamic stress intensity factors for multiple smooth cracks and hoop stress on cavities. Cavities are considered as closed curved cracks without singularity. Several examples are provided to investigate the effects of angular frequency of loading, characteristic length of defects and material properties on the dynamic fracture behavior of interacting cracks and cavities. The obtained results can be used to design coating-substrate structures under anti-plane time-harmonic loading.

Surface Waves on a Half-Space Due to a Time-Harmonic Loading on an Inertial Strip Foundation

An analytical approach has been applied to obtain the solution of Navier's equation for a homogeneous, isotropic half-space under an inertial foundation subjected to a time-harmonic loading. Displacement potentials were used to change the Navier's equation to a system of wave-type equations. Calculus of variation was employed to demonstrate the contribution of the foundation's inertial effects as boundary conditions. Use of the Fourier transformation method for the system of Poisson-type equations and applying the boundary conditions yielded the transformed surface displacement field. Direct contour integration has been employed to achieve the surface waves. In order to clarify the foundation's inertial effects, related coefficients were defined and a parametric study was conducted. Final results revealed that increasing the mass per unit length of the foundation or the frequency of the applied harmonic load intensifies the inertial effect factors. On the other hand, an increase in strip width or Poisson's ratio would imply reduction in the inertial effect factors.

The MLPG analyses of large deflections of magnetoelectroelastic plates

Abstract The von Karman plate theory of large deformations is applied to express the strains, which are then used in the constitutive equations for magnetoelectroelastic solids. The in-plane electric and magnetic fields can be ignored for plates. A quadratic variation of electric and magnetic potentials along the thickness direction of the plate is assumed. The number of unknown terms in the quadratic approximation is reduced, satisfying the Maxwell equations. Bending moments and shear forces are considered by the Reissner–Mindlin theory, and the original three-dimensional (3D) thick plate problem is reduced to a two-dimensional (2D) one. A meshless local Petrov–Galerkin (MLPG) method is applied to solve the governing equations derived based on the Reissner–Mindlin theory. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the centre of a circle surrounding it. The weak form on small subdomains with a Heaviside step function as the test function is applied to derive the local integral equations. After performing the spatial MLS approximation, a system of algebraic equations for certain nodal unknowns is obtained. Both stationary and time-harmonic loads are then analyzed numerically.