Anti-plane Shear Research Articles

Dynamic behavior of an inhomogeneous piezoelectric/piezomagnetic half space with a circular ring structure under SH wave

In this article, the dynamic characteristics of a circular ring structure in an inhomogeneous piezoelectric/piezomagnetic half spaces subjected to anti-plane shear wave is studied. The material parameters are assumed to have an exponential function distribution along the coordinate axis, and the Helmholtz equation includes variable coefficients due to inhomogeneity. First, by introducing new variables, the Helmholtz equation is transformed into standard form. Next, on the basis of boundary conditions, integral equations are setup and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor, electric field intensity concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

On the full dispersion Kadomtsev–Petviashvili equations for dispersive elastic waves

Full dispersive models of water waves, such as the Whitham equation and the full dispersion Kadomtsev–Petviashvili (KP) equation, are interesting from both the physical and mathematical points of view. This paper studies analogous full dispersive KP models of nonlinear elastic waves propagating in a nonlocal elastic medium. In particular we consider anti-plane shear elastic waves which are assumed to be small-amplitude long waves. We propose two different full dispersive extensions of the KP equation in the case of cubic nonlinearity and ”negative dispersion”. One of them is called the Whitham-type full dispersion KP equation and the other one is called the BBM-type full dispersion KP equation. Most of the existing KP-type equations in the literature are particular cases of our full dispersion KP equations. We also introduce the simplified models of the new proposed full dispersion KP equations by approximating the operators in the equations. We show that the line solitary wave solution of a simplified form of the Whitham-type full dispersion KP equation is linearly unstable to long-wavelength transverse disturbances if the propagation speed of the line solitary wave is greater than a certain value. A similar analysis for a simplified form of the BBM-type full dispersion KP equation does not provide a linear instability assessment.

Magnetoacoustic Wave Scattering and Dynamic Stress Concentration around the Elliptical Opening in Exponential-Gradient Piezomagnetic Materials

Based on the theory of magnetoacoustic coupled dynamics, the purpose of this paper is to evaluate the dynamic stress concentration near an elliptical opening in exponential-gradient piezomagnetic materials under the action of antiplane shear waves. By the wave function expansion, the solutions for the acoustic wave fields and magnetic fields can be obtained. Stress analysis is performed by the complex function method and the conformal mapping method, which are used to solve the boundary conditions problem, and is used to express the dynamic stress concentration coefficient (DSCC) theoretically. As cases, numerical results of DSCCs are plotted and discussed with different incident wave numbers and material parameters by numerical simulation. Compared with circular openings, elliptical openings are widely used in material processing techniques and are more difficult to solve. Numerical results show that the dynamic stress concentration coefficient at the elliptical opening is strongly dependent on various parameters, which indicates that the elliptical opening is more likely to cause crack and damage to exponential-gradient piezomagnetic materials.

Closed form solution for a semi-infinite crack moving in an infinite orthotropic material with a circular crack breaker under antiplane strain

This study investigates the influence of a circular crack breaker on mode-III deformation behavior of a semi-infinite crack in a homogeneous, elastic orthotropic material subjected to longitudinal shear loads. The Galilean transformation is employed to convert the governing wave equation to Laplace’s equation which is time independent, rendering the problem amenable to analysis within the realm of the classical theory of two-dimensional elasticity. Considering the geometrical configuration of the problem, the analytical solution of the problem is possible if the problem is transformed using the appropriate mapping function. Our construction of a holomorphic function that maps the circular hole into a straight line with the edge terminating at the origin is a novelty which enables the use of integral transform method to obtain an analytic solution of the displacement, leading to closed-form expression for mode-III stress intensity factor, K_{111}. The asymptotic values of the fields are obtained and shown to depend on the radius of the crack breaker. A parametric study shows that, for a fixed loading interval, a crack breaker of larger radius leads to increased stress intensity factor.

Micromechanics prediction of the effective elastic property of composites reinforced with cylindrically orthotropic nano fiber under antiplane shear

Micromechanics prediction of the effective elastic property of composites reinforced with cylindrically orthotropic nano fiber under antiplane shear

Adding multi-material regions embracing the tip leads to significant capacity increase in structures weakened by V-notches under antiplane shear and torsion

This study investigates how the insertion of multimaterial circular regions embracing the tip of a finite V-notch can be used to reduce the Notch Stress Intensity Factors (NSIFs) in structures subjected to antiplane shear or torsion. Towards this goal, this work presents a novel theoretical framework to calculate stress distributions and NSIFs in closed-form. Thanks to the new solution, it is shown that by tuning multimaterial region radii and elastic properties it is possible to significantly reduce the NSIFs and stress concentrations at the material interfaces.To investigate whether the proposed multimaterial system translates into increased structural capacity even in the presence of significant nonlinear deformations, computational simulations were conducted using nonlinear hyperelastic-damage and elasto-plastic-damage models. The preliminary results show increases of structural capacity up to 46% and of nominal strain at failure of up to 86% at the expenses of only a 8% reduction in structural stiffness.It is expected that a similar approach can be extended to other loading conditions (e.g. mode I and mode II, and fatigue) and that even larger gains can be obtained by performing thorough optimization studies.

Initial strength analysis of reinforced concrete monolithic supports for spherical and cylindrical mine workings

By using decomposition of hoop and radial components of the displacement vector to the trigonometrical and generalized power series, we obtained the new exact analytical solutions to problems of the equilibrium state of thick-walled heavy anisotropic central-and axial-symmetric bodies fixed on their external surfaces and subject to uniform internal lateral pressure. The estimation of an initial strength of cylindrical and spherical reinforced concrete monolithic mine working supports for a long-term storage of gas and oil is carried out on the basis of a multicriteria approach taking into account real damage mechanisms. They include damage from tension or compression in radial, hoop and axial directions, and from transversal and antiplane shear) of anisotropic axial- and central-symmetric bodies.

The effect of specimen thickness on the dislocation-free zone model of fracture

We use the method of continuously distributed dislocations to study the dislocation-free zone (DFZ) model of fracture in an infinite plate of finite thickness under externally applied uniform anti-plane shear stress. We formulate the equilibrium condition in terms of a set of two coupled singular integral equations using the representation of the total interaction force between two screw dislocations in a thin plate obtained by Eshelby and Stroh (1951). This set of singular integral equations is then solved numerically using the Gauss-Chebyshev integration formula resulting in the dislocation distribution functions in the crack and in the two plastic zones, the total number of screw dislocations in the plastic zone, the DFZ condition for a crack in a thin plate and the local mode III stress intensity factor at the crack tip.

Mode III analysis of a piezoelectric rectangular plane weakened by multiple cracks and cavities

This study presents an analytical model for a piezoelectric rectangular plane weakened by multiple cracks and cavities, under in-plane electric displacement and anti-plane shear stress. The defects are considered either electrically permeable or impermeable. First, the piezoelectric rectangular plane is weakened by a Volterra-type screw dislocation. The electric and stress displacement fields of the domain under consideration are determined using finite Fourier transform. Then, the problem is reduced into a set of Cauchy integral equations in the piezoelectric rectangular plane using the distributed dislocation approach. The solution to these integral equations determines electric displacement and stress intensity factors and hoop stress around cavities in the piezoelectric rectangular plane. Also, the boundary of the plastic zone is found using the stress field in the vicinity of crack tips and the Von Mises yield criterion. Finally, multiple numerical results are provided to illustrate the capability of the dislocation method in handling different cases of crack and cavity configurations and arrangements.

Crack front instability in mixed-mode I+III: The influence of non-singular stresses

In a previous paper (Leblond et al., 2011), a theoretical instability threshold was derived for the currently observed phenomenon of crack front fragmentation under mixed-mode I+III loading conditions. Instability modes were shown to emerge when the mode mixity ratio KIII0/KI0 exceeds some critical value [KIII0/KI0] that only depends on Poisson’s ratio. Unfortunately, the predicted threshold was found to be much larger than that observed in general. Numerical simulations of crack front fragmentation (Chen et al., 2015), based on a phase-field model, subsequently evidenced an important role of the specimen size on the non-coplanar instability. Here, we explore theoretically the influence of existence of some finite characteristic length(s), arising from the loading, by accounting for the presence of non-singular stresses Tij0 in the unperturbed configuration of the crack. By re-examining the linear stability analysis of Leblond et al. (2011) with these more general assumptions, we show that a negative non-singular stress Txx0 in the direction of propagation does not affect crack front fragmentation, while a negative non-singular Tzz0 in the direction of the crack front strongly hampers it. On the contrary, positive non-singular stresses Txx0 and Tzz0, or non-zero non-singular (antiplane shear) stresses Txz0, promote the fragmentation process, sometimes through the formation of facets that drift along the front as propagation proceeds. Large values of all three of these non-singular stresses may result in a significant lowering of the threshold value [KIII0/KI0] of the mode mixity ratio for instability, even possibly down to zero; which stresses out the potential existence of the fragmentation instability even in pure mode I. Yet, our results cannot explain the often observed formation of non-coplanar facets at extremely low mode mixity ratios; indeed the wavelength of the instability modes predicted are comparable to the finite characteristic length(s) introduced, of at least centimetric order of magnitude, while the formation of facets has been observed experimentally at scales as low as some tens of microns.

Unconventional cracks with dislocation-free zones

Using the method of continuously distributed dislocations, we study the dislocation-free zones for unconventional internal cracks in an infinite isotropic elastic medium under uniform remote anti-plane shear stress. The unconventional cracks are represented by a symmetric lip crack with two cusp tips and a symmetric airfoil crack with a single cusp tip. For each case, the equilibrium condition is formulated in terms of a single singular integral equation in the image plane. The resulting singular integral equation is solved numerically to arrive at the dislocation distribution function and the total number of dislocations in the plastic zone, the condition on remote loading and the local mode III stress intensity factor at the cusp tip. In the absence of the dislocation-free zones, the results are valid for the Dugdale model of these unconventional cracks.

Negative refraction for anti-plane elastic waves in canonical quasicrystalline laminates

Elastic anti-plane shear waves can be refracted negatively when they are transmitted across an interface between a homogeneous substrate and a transverse periodic laminate. To achieve pure negative refraction, the frequency of the source should be lower than the upper limit of the second transition zone of the harmonic spectrum of the laminate. An effective way to control the location of transition zones is to consider a canonical configuration for the laminate, a concept that originates from the properties of quasicrystalline sequences among which the Fibonacci one is a particular case. We give a detailed account of the classification in three families of canonical configurations and the role of canonical frequency. We exploit the knowledge of the scaling factor of the self-similar structure of the layout of transition zones for laminates of this kind to provide a quantitative tool to predict the relevant frequencies to accomplish negative refraction. We also investigate how the change of other parameters of the elementary cell may affect the values of those frequencies. The obtained results show that the features of quasicrystalline sequences may be profitably exploited for the realisation of elastic metamaterials.

Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics

In this paper, we present a center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders that is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution problem. Under suitable hypotheses, the resulting center manifold is finite dimensional and captures all sufficiently small bounded solutions. Compared with classical methods, the reduced ODE on the manifold is more directly related to the original physical problem and also easier to compute. The analysis is conducted directly in Hölder spaces, which is often desirable for elliptic equations. We then use this machinery to construct small bounded solutions to a variety of systems. These include heteroclinic and homoclinic solutions of the anti-plane shear problem from nonlinear elasticity; exact slow moving invasion fronts in a two-dimensional Fisher–KPP equation; and hydrodynamic bores with vorticity in a channel. The last example is particularly interesting in that we find solutions with critical layers and distinctive ‘half cat’s eye’ streamline patterns.

Anti-plane shear of a tunnel in an elastic half-space under surface loading: Analytic algorithm and approximate explicit solution

Anti-plane shear of a tunnel in an elastic half-space under surface loading: Analytic algorithm and approximate explicit solution

Homogenization of the wave equation with non-uniformly oscillating coefficients

The focus of our work is a dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g., functionally graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in , , within the framework of multiple scales expansion. When either or , this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in one dimension by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

Approximate controllability of nonsimple elastic plate with memory

<p style='text-indent:20px;'>In this paper, we give some qualitative results on the behavior of a nonsimple elastic plate with memory corresponding to anti-plane shear deformations. First we describe briefly the equations of the considered plate and then we study the well-posedness of the resulting problem. Secondly, we perform the spectral analysis, in particular, we establish conditions on the physical constants of the plate to guarantee the simplicity and the monotonicity of the roots of the characteristic equation. The spectral results are used to prove the exponential stability of the solutions at an optimal decay rate given by the physical constants. Then we present an approximate controllability result of the considered control problem. Finally, we give some numerical experiments based on the spectral method developed with implementation in MATLAB for one and two-dimensional problems.</p>

Second-order phase-field formulations for anisotropic brittle fracture

We address brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness. For these two classes, we develop two variational phase-field models based on the family of regularizations proposed by Focardi (2001), for which Γ-convergence results hold. Since both models are of second order, as opposed to the previously available fourth-order models for four-fold symmetric fracture toughness, they do not require basis functions of C1-continuity nor mixed variational principles for finite element discretization. For the four-fold symmetric formulation we show that the standard quadratic degradation function is unsuitable and devise a procedure to derive a suitable one. The performance of the new models is assessed via several numerical examples that simulate anisotropic fracture under anti-plane shear loading. For both formulations at fixed displacements (i.e. within an alternate minimization procedure), we also provide some existence and uniqueness results for the phase-field solution.

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

Higher order stress terms in sharp notch problems under pure‐out‐of‐plane loading

AbstractCurrent paper studies the effects of higher order terms (HOTs) on the stress and displacement fields around sharp V‐notched components under pure antiplane shear (mode III) loading. For this aim, two V‐notched test samples each containing four different notch angles are simulated in a finite element (FE) commercial software. Subsequently, the over‐deterministic technique is used as an approach to calculate the coefficients of singular and nonsingular terms out of the nodal displacement values. To verify the accuracy of the values calculated as the coefficients, the stress field reconstructed from the truncated series expansions is compared with the direct nodal outputs of FE analysis. The results demonstrate that the use of nonsingular terms of the asymptotic series expansion is necessary only in the case that the distribution of stress at far distances from the notch tip is sought and the notch angle is small. To check the stability of the results, some of the parameters which affect accurate determination of unknown coefficients and convergence of the numerical results are studied.

Lattice Boltzmann Method for Antiplane Shear with Non‐Mesh Conforming Boundary Conditions

AbstractLattice Boltzmann methods [1] have been extended beyond their initial usage in transport problems, and can be used to solve a broader range of partial differential equations, e.g. the wave equation [2]. Thereby they can be utilized for fracture mechanics [3]. In the context of antiplane shear deformation we previously examined a stationary crack [4, 5] with a finite width. In this work we present two implementation strategies for non‐mesh conforming boundary conditions, for which the bounding geometry does not need to adhere to the underlying lattice. This rectifies problems in modeling the crack. A numerical example shows the improvement compared to the previous results.