Two-dimensional Elastic Medium Research Articles

Propagation of coherent longitudinal and shear waves in two-dimensional elastic media with randomly distributed resonant cavities

Propagation of coherent elastic waves in solids containing randomly distributed resonant cylindrical cavities is studied numerically and experimentally. This work focuses especially on the case of water-filled cavities in aluminum alloys, which exhibit numerous Mie resonances in the frequency range investigated. Coherent wave measurements are performed in transmission at normal incidence for longitudinal waves and at oblique incidence beyond the critical angle for shear waves. Experimental observations are completed by numerical simulations made with an in-house code called MuScat adapted to the solid matrix case with wave conversions at the surface of each scatterer. The effective wave numbers obtained experimentally and numerically are compared with those predicted by statistical models based on the multiple scattering theory. Remarkable agreements between theoretical, numerical, and experimental results are obtained up to a concentration of 9.9% of cylinders.

A Stable Node-Based Smoothed Finite Element Method with Transparent Boundary Conditions for the Elastic Wave Scattering by Obstacles

In this work, a stable node-based smoothed finite element method with TBC (SNS-FEM-TBC) is proposed to solve the scattering of elastic plane waves by a two-dimensional (2D) homogeneous isotropic elastic medium. First, using Helmholtz decomposition, two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic scattering problem. Second, based on the analytical solutions of Helmholtz equations, TBC operators are deduced. Then, the gradient Taylor expansion is used to construct the stability term to deal with the instability of the original NS-FEM, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing domain as a circle. Finally, based on smoothed Galerkin weak formula, the SNS-FEM-TBC formula of linear algebraic system with linear smoothing gradient is derived. Numerical examples show that SNS-FEM can obtain more stable and accurate solutions than standard FEM. Moreover, the convergence rates of [Formula: see text] and [Formula: see text] semi-norm errors of SNS-FEM are faster.

Mechanical Deformation of Microtubules on a Two-Dimensional Elastic Medium.

Microtubule, the most rigid filamentous protein in cytoskeleton, plays significant roles in cellular mechano-transduction and mechano-regulation of cellular functions. In cells, the mechanical stress serves as a prevalent stimulus to frequently cause deformation of the microtubules participating in various cellular events. While the experimental and simulation-based approaches have confirmed the role of mechanical stress to tune mechanical properties of microtubule. Yet, the effect of mechanical force on the structural stability and the mechanism of microtubule deformation have remained obscure. Here, we describe the mechanical stress-induced deformation of microtubules using a custom-made mechanical device. We designed the device in a way which allows the microtubules to undergo deformation as response to the applied stress while attached on a two-dimensional elastic substrate through interaction with microtubule-associated motor protein, kinesin. We provide here the method to cause controlled bucking or fragmentation of microtubules by applying compressive or tensile stress on the microtubules, respectively. Such study is crucial to understand the mechanism of deformation in microtubules in cellular environment and their consequences in physiological activities.

Two-dimensional problem of two-temperature generalized thermoelasticity using memory-dependent heat transfer: an integral transform approach

We investigate the thermoelastic interactions in an isotropic and homogeneous perfectly conducting two-dimensional semi-infinite elastic medium. The bounding surface of the medium is assumed as stress free and subjected to a time-dependent thermal shock. A new two-temperature generalized thermoelasticity theory with memory-dependent derivative is employed for this study. The combined Laplace–Fourier transforms are applied to solve the non-dimensional governing equations to find the solutions for the field variables in the transform domain. An application is considered to enable us to get complete solutions. Numerical results of the field variables are presented graphically to discuss the effect of various parameters of interest.

An augmented time reversal method for source and scatterer identification

We propose a computational method which improves the identification of sources and scatterers in a two-dimensional elastic medium using the Time Reversal (TR) technique. The identification procedure makes use of partial numerical data available at a small number of sensors. While in realistic applications these data are obtained by lab or field measurements, in the present work they are synthesized by solving the corresponding forward problem. We introduce an “augmentation” procedure that is utilized in addition to the TR technique and investigate its benefits. The augmentation procedure involves the solution of a suitable elliptic problem at selected time steps. In practice, one needs to solve, as a pre-process, a small number of “fundamental” elliptic problems which is equal to the number of measured quantities. A version of the augmentation procedure, which is easier to implement and involves almost no computational cost, is also presented. The local elastic energy in the domain is employed to decide whether the TR solution has properly refocused at the source. Cost functionals, whose minima are sought, are constructed for the source and scatterer identification problems. We conclude that augmentation improves the identification performance, in particular in the presence of noisy measurements. In some cases augmented TR succeeds in the identification whereas non-augmented TR fails.

A fast biem algorithm to evaluate seismic ground response in 2d elastic time domain

Since the topography and surface irregularities of the terrain have a significant effect on the seismic response of the earth's surface, the numerical analysis using boundary element method in elastodynamics, which leads to a significant increase in the number of DOF and stiffness matrix dimension, as well as the sparse and unsymmetrical structure of stiffness matrix, indicates the inefficiency of the conventional method. This paper presents a fast boundary element algorithm in seismic analysis of two-dimensional elastic media for the first time. In the proposed method, instead of the usual node-to-node, a method of cell-to-cell relation method is implemented by hierarchy tree structure. In addition, the Plane Wave Time Domain algorithm and the Iterative method has been used to solve a system of equations which increases the speed of network convergence, especially in high degrees of freedom that has not been used in the study of the seismic waves’ response yet. Nowadays, this new algorithm adopts for investigating the response wave fields of electrical, magnet [1], acoustic and thermal conductivity which are discussed in details in various articles and books.

Calculation and Analysis of Interaction Between Buried Pipeline and Soil

In order to gain a clear idea of mechanism of stress and deformation for buried pipeline, the soil pressure on the buried pipeline computation boiled down to solving the gravity field in two-dimensional elastic medium embedded with an elastic ring hole stress concentration problem. Based on considering the deformation coordination of pipe-soil contact surface, the stress function method was used to derive the elasticity solution of the soil pressure around the pipeline. It established the analytical relationship between pressure around the pipeline and pipeline section stiffness, providing a new way for pipeline force calculating. Example analysis indicated that the pipe circumferential stress using the stress function method was quite closed to the finite element calculation and the simplified stress distribution around pipeline was convex parabola. Through the calculation of pipe-soil interaction, it was further indicated that besides the positive soil stress acting on the buried pipeline, there had large value of shear stress along the pipeline. The role of this part should be considered in calculating the force acting on the buried pipeline.

Buckling of microtubules on elastic media via breakable bonds

Buckling of microtubules observed in cells has been reconstructed on a two-dimensional elastic medium consisting of kinesins grafted over compressible substrates, enabling precise control of experimental conditions and quantitative analysis. However, interpretations of the observations have ambiguities due to inevitable experimental difficulties. In this study, with computer simulations, we investigated importance of the mode of interaction of microtubule with elastic medium in the buckling behavior of microtubule. By taking into consideration of forced-induced detachments of kinesins from microtubules, our simulations reproduced the previous experimental results, and showed deviations from predictions of the elastic foundation model. On the other hand, with hypothetical linkers permanently bound to microtubules, our simulation reproduced the predictions of the elastic foundation model. By analyzing the results of the simulations, we investigated as to why the difference arose. These findings indicate the importance of the mode of interaction of microtubule with the medium in the buckling behavior of microtubule. Our findings would bring new insights on buckling of microtubules in living cells.

Buckling of Microtubules on a 2D Elastic Medium.

We have demonstrated compression stress induced mechanical deformation of microtubules (MTs) on a two-dimensional elastic medium and investigated the role of compression strain, strain rate, and a MT-associated protein in the deformation of MTs. We show that MTs, supported on a two-dimensional substrate by a MT-associated protein kinesin, undergo buckling when they are subjected to compression stress. Compression strain strongly affects the extent of buckling, although compression rate has no substantial effect on the buckling of MTs. Most importantly, the density of kinesin is found to play the key role in determining the buckling mode of MTs. We have made a comparison between our experimental results and the ‘elastic foundation model’ that theoretically predicts the buckling behavior of MTs and its connection to MT-associated proteins. Taking into consideration the role of kinesin in altering the mechanical property of MTs, we are able to explain the buckling behavior of MTs by the elastic foundation model. This work will help understand the buckling mechanism of MTs and its connection to MT-associated proteins or surrounding medium, and consequently will aid in obtaining a meticulous scenario of the compression stress induced deformation of MTs in cells.

G0300601 円形介在物内に別種の偏心円形介在物が存在する弾性体の応力解析

This paper presents the theoretical solution of a two-dimensional isotropic elastic medium (matrix) with eccentric two circular inclusions (ring model) under the in-plane problems. Inner inclusion is perfectly bonded to the outer inclusion. These inclusions have different elastic moduli, radii and central points. The matrix is infinite extent and subjected to arbitrary in-plane loading, for example, uniform stresses at infinity, as well as a concentrated force at an arbitrary point. These inclusion problems have many applications in disciplines such as engineering. The purpose of the present study is to apply the reflection principle and produces a general solution. Using these solutions, several numerical examples are presented with graphs.

Scale-dependent homogenization of random composites as micropolar continua

Abstract A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill–Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ɛ; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.

Two-dimensional polynomial eigenstrain formulation of boundary integral equation with numerical verification

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.

Cohesive Interface Modeling of Debonding Failure in FRP Strengthened Beams

A theoretical model that incorporates the concept of the cohesive interface approach for the debonding analysis of reinforced concrete beams strengthened with externally bonded fiber reinforced polymer (FRP) strips is presented. The cohesive interface concept is adopted for modeling of the debonding process near the critical adhesive-concrete interface, whereas the adhesive layer itself is modeled as a two-dimensional elastic medium. Thus, the stress and deformation fields within the adhesive layer, the coupling between the shear and normal stresses and, especially, their influence on the tractions across the cohesive interface are taken into account. The nonlinear relations between the tractions and the displacement jumps across the cohesive interface are derived using a potential function and account for the peeling effects and for the coupling between the shear-slip and the peeling-separation laws. Numerical results that examine the capabilities of the model, provide insight into the stability characteristics of the debonding mechanism, and highlight some aspects of the debonding problem are presented. A summary and conclusions close the paper.

Boundary Perturbation Solution for Nearly Circular Holes and Rigid Inclusions in an Infinite Elastic Medium

The boundary perturbation method is used to solve the problem of a nearly circular rigid inclusion in a two-dimensional elastic medium subjected to hydrostatic stress at infinity. The solution is taken to the fourth order in the small parameter epsilon that quantifies the magnitude of the variation of the radius of the inclusion. This result is then used to find the effective bulk modulus of a body that contains a dilute concentration of such inclusions. The corresponding results for a cavity are obtained by setting the Muskhelishvili coefficient κ equal to −1, as specified by the Dundurs correspondence principle. The results for nearly circular pores can be expressed in terms of the pore compressibility. The pore compressibilities given by the perturbation solution are tested against numerical values obtained using the boundary element method, and are shown to have good accuracy over a substantial range of roughness values.

Scattering attenuation, dispersion and reflection ofSHwaves in two-dimensional elastic media with densely distributed cracks

We compute the synthetic seismograms of multiply scattered SH waves in 2-D elastic media with densely distributed parallel cracks. We assume randomly distributed cracks in a rectangular-bounded region, which simulate a cracked zone. The crack surfaces are assumed to be stress-free. When the incident wavelength is longer than the crack size, the delay in the arrival of the primary wave is observed at stations beyond the cracked zone and the amplitude of the primary wave is amplified in the cracked zone in the synthetic seismograms. This is because the cracked zone behaves as a low velocity and soft material to the incident long-wavelength wave due to the crack distribution. When the half-wavelength of the incident wave is shorter than the crack length, the scattered waves are clearly observed in the synthetic seismograms and the amplitude of the primary wave is largely attenuated beyond the cracked zone. The calculated attenuation coefficient Q−1 of the primary wave is directly proportional to the crack density in the range of νa2≤ 0.1, where ν and a are the number density and half the length of cracks, respectively. This is consistent with that obtained by a stochastic analysis based on Foldy's approximation. A periodic distribution of cracks in a zone is considered as an utterly different model in order to investigate the effect of spatial distributions on the attenuation and dispersion of seismic waves. When cracks are distributed densely, the values of Q−1 for the periodic crack distribution appear to differ from those for the random distribution of cracks in the low wavenumber range. This suggests that the effect of multiple interactions among densely distributed cracks depends on not only the density but also the spatial distribution of cracks at low wavenumbers. The calculated phase velocity of the primary wave is consistent with that from the stochastic analysis in the range of νa2≤ 0.1 and does not depend on the spatial distribution of cracks. This suggests that the multiple crack interactions have a smaller effect to the phase velocity. Therefore, the crack density can be estimated from the values of the phase velocity for the cases of densely distributed cracks even if the effect of the multiple crack interactions is not considered. We can clearly observe the reflected waves in the synthetic seismograms. The elastic constant of a single anisotropic layer equivalent to the cracked zone is derived from the crack density at the long-wavelength limit. The reflection coefficients calculated from the synthetic seismograms are consistent with those of the anisotropic layer calculated from its elastic constants and thickness in low wavenumber range. This means that a fracture zone distributed parallel cracks is considered as an anisotropic layer for long incident wavelengths. Therefore, the elastic constants, crack density and the thickness of the fracture zone can be estimated from the frequency dependence of the reflection coefficients for long incident wavelengths. On the contrary the wavenumber dependence of the reflection coefficients cannot be explained theoretically in the high wavenumber range.

Elastic wave propagation through a distribution of dislocations

We study the coherent propagation of an elastic wave in a two-dimensional continuous elastic medium filled with dislocation arrays randomly distributed and oriented in space. This configuration reasonably mimics grain boundaries in polycrystals. Interest is in evaluating the plastic contributions to the multiple scattering of waves in polycrystals that may superpose to other known scattering processes, like scattering due to inhomogeneities of elastic properties among grains. Calculations are performed in a multiple scattering formalism, based on the derivation of the so-called mass operator, in the approximation of weak scattering. We find that sound attenuation increases when the frequency decreases, a trend opposite to the usual behavior, suggesting that dislocations could sensibly modify the acoustic properties of materials at low frequency.

Stress field around arbitrarily shaped cracks in two-dimensional elastic materials.

The calculation of the stress field around an arbitrarily shaped crack in an infinite two-dimensional elastic medium is a mathematically daunting problem. With the exception of few exactly soluble crack shapes the available results are based on either perturbative approaches or on combinations of analytic and numerical techniques. We present here a general solution of this problem for any arbitrary crack. Along the way we develop a method to compute the conformal map from the exterior of a circle to the exterior of a line of arbitrary shape, offering it as a superior alternative to the classical Schwartz-Cristoffel transformation. Our calculation results in an accurate estimate of the full stress field and in particular of the stress intensity factors K(I) and K(II) and the T-stress which are essential in the theory of fracture.

円形介在物内の任意点に特異荷重が作用する面外弾性体問題の解析

In this paper, the theoretical solutions are shown for the cases of a two-dimensional isotropic elastic medium containing an isotropic elastic circular inclusion loaded in singular disturbances at a finite point inside and having different isotropic elastic circular inclusions, under the anti-plane loading. It is assumed that the boundary conditions among the matrix and the elastic inclusion are perfectly bonded. This analysis is based on the complex variable method using the Mebius Transformation. Using these techniques, several examples are shown by graphical representation.

High-Order Analysis of Unidirectional Sandwich Panels with Flat and Generally Curved Faces and a “Soft” Core

The bending behavior of a unidirectional sandwich panel with flat and generally curved faces and a flexible core in the vertical direction is investigated. The studied panels consist of an upper flat face sheet, a core of variable thickness, and a lower face sheet that can take any geometrical layout described by an analytical function. The core is assumed a two-dimensional elastic medium with shear and transverse vertical rigidities only, and the face sheets are considered as membrane and bending members made of metallic or composite materials. The term unidirectional refers here either to a beam (narrow panel) or to a wide beam (panel) undergoing a cylindrical bending. The field equations and the boundary conditions are derived via the variational principle of the virtual work and transformed from the local polar coordinate system of the curved face into the global Cartesian one. Higher order effects due to flexibility of the core in the form of nonlinear deformation fields are incorporated in the proposed analysis as a result of the closed-form solution of the field equations and are not postulated a priori. Numerical results in terms of deformations, stresses, and stress resultants are presented for some typical cases of flat-curved panels. The effect of a wide range of layouts, various boundary conditions, and the influence of manufacturing imperfections in the form of initially wrinkled face are investigated. The results demonstrate the capabilities and generality of the proposed model and present its ability to predict the high-order localized effects that characterize such panels.