Anti-plane Problem Research Articles

Антиплоские задачи о движении осциллирующей нагрузки по границе упругой изотропной полосы при наличии поверхностных напряжений

In this paper, symmetric and antisymmetric antiplane problems about the motion with a constant subsonic velocity of an oscillating load along the boundary of an elastic isotropic nanothin strip are considered. The nanoscale strip thickness is considered by introducing surface stresses in accordance with the Gurtin-Murdoch theory. According to this theory, it is assumed that, in addition to external loads, surface stresses act on the layer boundaries, which are described by Hooke's surface law. As a result, the properties of the elastic material of the strip with nanoscale thickness become different from the material properties of a regular-sized body. A standard technique was used for the so-lution, including the application of limiting absorption principle, the Fourier transform over infinitely extended co-ordinate and the theory of residues for finding the inverse Fourier transform. For various strip thicknesses, solutions were obtained in the form of series in natural waves, dispersion relations were studied, and graphs of the displace-ment amplitudes along the thickness were plotted. The analysis showed that for fixed values of frequency and velocity of the source, the values of non-negative real wave numbers are greater in the presence of surface stresses than the values of the wave numbers for the classical case of the problem without surface stresses. It is noted that surface stresses have a significant effect only when the strip thickness decreases to nanosizes.

Stress effects on electric currents in antiplane problems of piezoelectric semiconductors over a rectangular domain

Stress effects on electric currents in antiplane problems of piezoelectric semiconductors over a rectangular domain

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.

On integral and differential formulations in nonlocal elasticity

The paper is concerned with comparative analysis of differential and integral formulations for boundary value problems in nonlocal elasticity. For the sake of simplicity, the focus is on an antiplane problem for a half-space with prescribed shear stress along the surface. In addition, 1D exponential kernel depending on the vertical coordinate is considered.First, a surface loading in the form of a travelling harmonic wave is studied. This provides a counter-example, revealing that within the framework of Eringen’s theory the solution to the differential model does not satisfy the equation of motion in nonlocal stresses underlying the related integral formulation.A more general differential setup, starting from singularly perturbed equations expressing the local stresses through the nonlocal ones, is also investigated. It is emphasised that the transformation of the original integral formulation to the differential one in question is only possible provided that two additional conditions on nonlocal stresses hold on the surface. As a result, the formulated problem subject to three boundary conditions appears to be ill-posed, in line with earlier observations for equilibrium of a nonlocal cantilever beam.Next, the asymptotic solution of the singularly perturbed problem, subject to a prescribed stress on the boundary, together with only one of the aforementioned extra conditions, is obtained at a small internal size. Such simplification may be justified when only one of the stresses demonstrates nonlocal behaviour; a similar assumption has been recently made within the so-called dilatational gradient elasticity. Three-term expansion is obtained, leading to a boundary value problem in local stresses over interior domain. The associated differential equations are identical to those proposed by Eringen, however, the derived effective boundary condition incorporates the effect of a nonlocal boundary layer which has previously been ignored. Moreover, the calculated nonlocal correction to the classical antiplane problem for an elastic half-space, coming from the boundary conditions is by order of magnitude greater than that appearing in the equations of motion. Finally, it is shown that the proposed effective condition supports an antiplane surface wave.

The Anti-Plane Problem of Collinear Interface Cracks Emanating From a Circular Hole in 1D Hexagonal Quasicrystal Bi-Materials

The Anti-Plane Problem of Collinear Interface Cracks Emanating From a Circular Hole in 1D Hexagonal Quasicrystal Bi-Materials

Fundamental formulation for anti-plane eigenstrain problems

In this paper, the anti-plane eigenstrain problems are addressed in the framework of plane strain. A fundamental solution for an anti-plane eigenstrain nucleus located in an infinite plane is derived first. With this solution, the anti-plane eigenstrain toughening problems are formulated by Green's function method, which includes plastic strain shielding problems and transformation toughening problems. Also, with the equivalent eigenstrain principle and the Green's function method, its application to formulating inhomogeneous inclusion problems subjected to anti-plane load is demonstrated. The fundamental formulations and the related examples introduced here may pave a way to explore more complex problems in the corresponding aspects.

A new approach to solve the anti-plane crack problems by the method of fundamental solutions

This paper expands the method of fundamental solutions (MFS) to solve the anti-plane crack problems based on the traditional fundamental solution by the conformal mapping technique without using the domain decomposition technique and/or the crack Green's function. Based on the complex analysis and the conformal mapping technique, the approximate solution of the complex analytic function is derived for the MFS. The modified boundary conditions are deduced to construct the interpolation equations of the MFS. The conformal mapping technique naturally introduces singularities of the stress fields and the stress intensity factors (SIFs) for the anti-plane crack problems without using the crack Green's function and the domain-decomposition technique. The obtained numerical examples illustrate the validity and accuracy for the anti-plane crack problems. The proposed method is an expansion of the traditional MFS, and can be applied for many anti-plane elastic problems with the simple concept and the easy numerical implementation.

Spectral formulation of the boundary integral equation method for antiplane problems

A spectral formulation of the boundary integral equation method for antiplane problems is presented. The boundary integral equation method relates the slip and the shear stress at an interface between two half-planes. It involves evaluating a space-time convolution of the shear stress or the slip at the interface. In the spectral formulation, the convolution with respect to the spatial coordinate is performed in the spectral domain. This leads to greater numerical efficiency. Prior work on the spectral formulation of the boundary integral equation method has performed the elastodynamic convolution of the slip at the interface. In the present work, the convolution is performed of the shear stress at the interface. The spectral formulation is developed both for an interface between identical solids and for a bi-material interface. It is validated by numerically calculating the response of the interface to harmonic and to impulsive disturbances and comparing with known analytical solutions. To illustrate use of the method, dynamic slip rupture propagation with a slip-weakening friction law is simulated.

Microscale size effects in piezomagnetic material for the anti-plane problem

A continuum model of piezomagnetic material with strain, magnetic and piezomagnetic coupling gradient effects is proposed using a variational principle in this work. This model is employed to an anti-plane problem, and a general solution is constructed in polar coordinates. Special attention is paid to microstructural effects on the magnetic and mechanical response in an infinite piezomagnetic medium with a void. It is found that the microstructural length scales have a significant influence on the mechanic and magnetic fields. The three length scales (corresponding to strain, magnetic and piezomagnetic coupling gradients) are indispensable to describe the nonlocal effects of piezomagnetism. Additionally, controlling the direction and magnitude of the magnetic field at the edge of the void can be achieved by adjusting the microstructural length scales of the piezomagnetic medium.

Jordan-form special solutions corresponding to arbitrary anti-plane shear forces in polynomial forms imposed on bimaterial interfacial crack surfaces

Bimaterial anti-plane interfacial crack problems with arbitrary distributed forces applied on crack surfaces are numerically investigated. The problems are in essence a kind of boundary value problems about Laplace's equations. Previous work of using Symplectic analytical singular element (SASE) with high-order accuracy and other singular finite elements for calculating mode III stress intensity becomes invalid when the distributed forces on each surface are different. In this paper, Jordan-form special solutions (JFSS) are proposed for the first time. We derive the JFSS for present problems and use them to improve displacement modes of SASE. The distributed forces are therefore converted into equivalent nodal forces of the nodes around SASE. The new SASE is found to be directly and conveniently used for cases where the two distributed forces are either equal (even both zero) or not equal. Numerical examples are provided to demonstrate the effectiveness and accuracy of the present method. The necessity and effectiveness of the JFSS is also validated. Moreover, the JFSS can be applied to some other physical problems represented by the Laplace's equation.

The propagation of transient waves in two-dimensional square lattices

The aim of this article is to study the attenuation of transient low-frequency waves in 2D lattices in both plane and antiplane problems. The main idea of this article is that analytical solutions to problems of mechanics of discrete periodic media can be obtained by a method of asymptotic inversion of the Laplace and Fourier transforms in the vicinity of the quasi-front of infinitely long waves; moreover, in this method it is possible to take into account the contribution of short waves. Using this method, we obtain asymptotics of perturbations in lattices in plane and antiplane formulations under a local transient load. Besides, we show that equations describing 2D plane motion of a square lattice can be represented in the form of two linearly independent wave equations, each of which contains one unknown function only. By analogy with the theory of elasticity, one equation describes the propagation of shear waves in the lattice, while the other equation describes the propagation of longitudinal waves. As a result, it is shown that, in a homogeneous infinite lattice, a load can be specified in such a manner that either predominantly longitudinal or predominantly shear waves are formed. The problems under study are also solved by a finite difference method. The qualitative and quantitative correspondence of asymptotic and numerical solutions is shown.

Applications of the Trefftz method to the anti-plane fracture of 1D hexagonal piezoelectric quasicrystals

In terms of solving fracture mechanics problems of quasicrystals, the boundary element method known as Trefftz Method is used for the first time. In this paper, the collocation Trefftz method as an indirect Trefftz method has been demonstrated to be an excellent numerical method for the anti-plane fracture problems in quasicrystals plate, which can exactly satisfy the governing equations and approximately satisfy the boundary conditions. The algorithms of the collocation Trefftz method and the trial functions for the anti-plane fracture problems of piezoelectric quasicrystals are presented. The effects of the geometry size, the loadings and the materials properties on the fracture results are studied. For the materials with three dimensional constitutive relations, such as piezoelectric quasicrystals, the Trefftz method is an effective numerical method to solve the fracture problems.

Dynamic anti-plane behavior of rare earth giant magnetostrictive medium with a circular cavity defect in semi-space

An analytical solution to the anti-plane dynamics problem of semi-space rare earth giant magnetostrictive media with circular cavity defects near the horizontal boundary under the action of SH wave is studied. Based on the Helmholtz theorem and the theory of complex function, the elastic-magnetic dynamic equation of magnetostrictive medium is established, and the semi-space incident wave field is written. In addition, based on the theory of complex function and the method of wave function expansion, the expression of the wave function of the scattered displacement field and the corresponding magnetic potential of the scattered wave under the condition of no stress and magnetic insulation of the horizontal boundary are obtained. Then, based on the conditions of free boundary stress, continuous magnetic induction intensity and continuous magnetic potential around the circular cavity, the infinite linear algebraic equations are established. Finally, the analytical expressions of dynamic stress concentration factor and magnetic field intensity concentration factor around circular cavity in semi-space rare earth giant magnetostrictive medium are obtained. Numerical examples show that the analysis results depend on the following parameters: permeability, dimensional-piezomagnetic coefficient, frequency of the incident wave, incident angle, distance between the circular cavity and horizontal boundary. These results have certain reference value for the study of non-destructive testing and failure analysis of rare earth giant magnetostrictive materials.

Modeling of deformation of the bimaterial with thin Non-linear interface inclusion

An incremental approach to solving the antiplane problem for bimaterial media with a thin, physically nonlinear inclusion placed on the materials interface is discussed. Using the jump functions method and the coupling problem of boundary values of the analytical functions method we reduce the problem to the system of singular integral equations (SSIE) on jump functions with variable coefficients allowing us to describe any quasi-static loads (monotonous or not) and their influence on the stress-strain state in the bulk. To solve the SSIE problem, an iterative analytical-numerical method is offered for various non-linear deformation models. Numerical calculations are carried out for different values of non-linearity characteristic parameters for the inclusion material. Their parameters are analyzed for a deformed body under a load of a balanced concentrated force system.

Simulation of antiplane shear problems with multiple inclusions using the generalized finite difference method

In this paper, the generalized finite difference method is adopted for solving antiplane shear problems with multiple inclusions. This method is applied to domains in matrix and inclusions respectively in which the local Taylor expansion and moving least square method are applied to form the local system. And the final sparse system is obtained through satisfying the continuity and equilibrium conditions along the interface between matrix and inclusions. Two numerical examples containing several number of inclusions with various locations, shapes, sizes as well as different shear moduli are provided to assess the applicability and accuracy of the proposed method. It is verified that the proposed method can be used for problems when the ratio of shear moduli is extremely large. The comparison is made between the proposed method and reference solutions.

Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals

To effectively reduce the field concentration around a hole or crack, an anti-plane shear problem of a nano-elliptical hole or a nano-crack pasting a reinforcement layer in a one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) is investigated subject to remotely mechanical and electrical loadings. The surface effect and dielectric characteristics inside the hole are considered for actuality. By utilizing the technique of conformal mapping and the complex variable method, the phonon stresses, phason stresses, and electric displacements in the matrix and reinforcement layer are exactly derived under both electrically permeable and impermeable boundary conditions. Three size-dependent field intensity factors near the nano-crack tip are further obtained when the nano-elliptical hole is reduced to the nano-crack. Numerical examples are illustrated to show the effects of material properties of the surface layer and reinforced layer, the aspect ratio of the hole, and the thickness of the reinforcing layer on the field concentration of the nano-elliptical hole and the field intensity factors near the nano-crack tip. The results indicate that the properties of the surface layer and reinforcement layer and the electrical boundary conditions have great effects on the field concentration of the nano-hole and nano-crack, which are useful for optimizing and designing the microdevices by PQC nanocomposites in engineering practice.

Nonlinear elastic constitutive modeling of [formula omitted]-Ge

We present a constitutive framework for modeling the nonlinear elastic behavior of α-Germanium (Ge). Starting with all possible phase changes Ge sustains under compression, we correspond to each space group of every phase the arithmetic symmetry group by viewing Ge as a multilattice. We then focus on the mother α phase of Ge. Confining ourselves to weak transformation neighborhoods and adopting the Cauchy–Born rule we work with the classical symmetries of continuum mechanics. Since there is no available representation theory for the symmetry group of α-Ge, we use integrity basis theory to find the most general expression for the arguments of the energy. Using energy’s expression we evaluate the stress tensor in its most general form. We then study two problems of increasing difficulty: simple tension/compression and anti-plane shear. For the first class of problems we give closed form solutions, while the anti-plane shear problem complicates the equations dramatically; we give conditions for the solvability of the field equations ruling the shift vector.

Antiplane shear crack in a functionally graded material strip with surface elasticity

When the dimension of a structure falls to the micro-/nanoscale, surface effect is significant and plays a key role in affecting the mechanical behavior. This article studies the influence of surface elasticity on the stress intensity factor of an antiplane shear crack embedded in an elastic strip made of functionally graded materials. Surface elasticity is applied on the strip surfaces and crack faces, and classic elasticity is invoked for the strip interior. An antiplane shear crack problem is solved for a symmetric FGM with a crack parallel to the strip surfaces. The associated problem is converted to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces through the Fourier transform and then to a singular integro-differential equation with Cauchy kernel. The Galerkin method is applied to expand the crack face displacement as a Chebyshev series, and the singular integro-differential equation reduces to a system of algebraic linear equations. Stress intensity factors at the crack tips and the out-of-plane displacement on the crack faces are calculated numerically. It is found that surface elasticity and gradient index strongly alter the bulk stress and its intensity factors near the crack tips. Positive surface shear modulus decreases the mode III stress intensity factors and negative surface shear modulus has an opposite behavior. The influence of the variation of material gradient on the mode III stress intensity factors is expounded in graph.

Anti-plane seismic performance of a shallow-buried tunnel with imperfect interface in anisotropic half-space

The current research presents a semi-analytical solution for the anti-plane dynamic response of a shallow-buried lined tunnel. The tunnel with an imperfect interface between the tunnel lining and its surrounding medium is embedded in an anisotropic and elastic half-space medium. For anti-plane problems, three elastic constants can describe the characteristics of medium anisotropy according to the generalized Hooke’s Law. Therefore, the complex variable function method and multi-polar coordinate system were applied to express the wave fields with three unknown constants in this study. These constants also automatically satisfy the zero-stress boundary condition on the horizontal ground surface in the half-space and the Sommerfeld radiation condition at infinity. The imperfect interface is represented by a series of elastic spring model. Meanwhile, parametric sensitivity analyses indicate that both the medium anisotropy and the imperfect interface stiffness affect significantly the dynamic response of the medium-lining system including stress and ground surface displacement. Besides, the influence of both factors is also dependent on incidence direction and frequency.

Anti-plane problem of nanocrack with surface piezoelectricity—a finite-form solution

A solution in finite form for the screw dislocation interacting with the nanocrack incorporating surface piezoelectricity under anti-plane loads is developed. By the boundary integration and the Cauchy’s integral formula to convert the problem into a first-order differential equation, the finite-form solution in physical region is established via the mapping technique. At the crack tip, the results from the present solution indicate that the electric displacements and the stresses are finite and the singularities are removed when the surface piezoelectricity is incorporated, which is against those in the classic solution and also different from other reported solutions where the singularities are weakened but non-trivial. The numerical results shows that the stresses, the electric displacements and the dislocation forces are strongly size dependent and their magnitudes reduce with the decrease in crack length, and the results approach to those by the classic solution with an increasing in the crack length.