Anisotropic Elastic Half-space Research Articles

ПЕРВАЯ КРАЕВАЯ ЗАДАЧА ДИНАМИКИ АНИЗОТРОПНОГО УПРУГОГО ПОЛУПРОСТРАНСТВА ПРИ ДЕЙСТВИИ ДОЗВУКОВЫХ ТРАНСПОРТНЫХ НАГРУЗОК

The first boundary value problem of the theory of elasticity for an anisotropic elastic half-space is solved when a transport load moves along its surface. The subsonic Raleigh case is considered, when the velocity of motion is less than the velocity of propagation of bulk and surface elastic waves. The Green’s tensor of the transport boundary value problem is constructed and on its basis the solution of boundary value problems for a wide class of distributed traffic loads is given. To solve the problem, the methods of tensor and linear algebra, integral Fourier transform, and operator method for solving systems of differential equations were used. The obtained solution makes it possible to investigate the dynamics of the rock mass for a wide class of transport loads, in a wide range of velocities, both low velocities and high velocities, and to evaluate the strength properties of the rock mass under the influence of road transport. In particular, determine the permissible velocities of its movement and carrying capacity. In addition, a investigation on its basis of the movement of the day surface along the route will make it possible to establish criteria for the seismic resistance of ground structures and the permissible distances of their location from the route.

Scattering and Backscattering Study of Mechanical Plane Wave in Composite Materials Plates (Earth model 1066B and LiNbO3)

Reflection and refraction phenomenon pattern of elastic plane wave at the interface between anisotropic monoclinic elastic half-space and isotropic elastic half-spaces is studied. Closed-form expression for phase velocity is obtained. Reflection and transmission coefficients are obtained using the method of Cramer's rule in determinant form. Also, the energy ratios are calculated in terms of reflection and transmission coefficients. Numerical examples are considered to exhibit all the findings graphically. The energy conservation law is implemented at each angle of incidence to validate the numerical results, and it is found that energy ratios are in good agreement with the energy conservation law.

Reduced model for the surface dynamics of a generally anisotropic elastic half-space

Near-surface resonance phenomena often arise in semi-infinite solids. For instance, when a moving load with a speed v close to the surface wave speed v R is applied to the surface of an elastic half-space, it will give rise to a large-amplitude disturbance inversely proportional to v − v R . The latter can be determined by a multiple-scale approach using an extra slow time variable. It has also been shown for isotropic elastic half-spaces that the reduced governing equation thus derived is capable of describing the surface wave contribution even for arbitrary dynamic loading. In this paper, we first derive the analogous evolution equation for a generally anisotropic elastic half-space, and then assess its applicability in the study of travelling waves in a half-space that is coated with a continuous array of spring-like vertical resonators or bonded to an elastic layer of different properties. Our results are validated by comparison with previously known results, and illustrative calculations are carried out for a fibre-reinforced half-space and a coated half-space that is subjected to a finite deformation.

Boundary element analysis of dynamic interaction of anisotropic elastic half-space and structure under impact load

In this paper, a direct boundary element formulation for numerical modelling of threedimensional dynamic soil-structure interaction is briefly described. Formulation is applicable to problems involving isotropic and anisotropic elastic materials. It is based on the Laplace domain regularized boundary integral equations for the displacements under assumption of vanishing initial conditions and zero body forces. Mixed boundary elements are employed for the spatial discretization of the boundary integral equations. Subdomain technique is used to tackle problems involving piecewise homogeneous material properties. The described formulation is employed for the boundary element analysis of dynamic soil-structure interaction problem involving a structure which contains a cavity with a complex geometrical form. Soil is represented by a transversely isotropic elastic half-space. Effect of material anisotropy of half-space on the transient response of the embedded structure is assessed. It is observed that the shear modulus anisotropy have substantial impact on the displacements response. Influence of the Young’s modulus in the plane of isotropy is lesser but still pronounced.

Elastic field by a dislocation loop in an anisotropic elastic half-space with general boundary conditions and its application in nanoindentation of single crystals

The elastic displacement and stress fields induced by a dislocation loop of polygonal shape within an anisotropic homogeneous half-space with general boundary conditions on the flat surface z = 0 are obtained by employing the corresponding point-force Green's functions. We first study in detail the effects of the eight sets of boundary conditions on the elastic fields of dislocation loops. We then identify the elastic fields of dislocation loops corresponding to three of the eight sets of boundary conditions. These are the traction-free, completely, and partially sticking contact between the indenter and the indented single crystals, which are employed to establish the distribution dislocation loop model for simulating nanoindentation of single crystals. It is the first time that significant differences in the magnitude of sink-in and pile-up of the indented profile, and of the residual stress components, corresponding to different frictional boundary conditions, are observed, which could serve as benchmark in future nanoindentation study.

BEM modeling of a 3D homogeneous anisotropic elastic half space under dynamic load

In this paper, we consider a problem of a homogeneous anisotropic and linearly elastic half space subjected to dynamic loading. Zero body forces and vanishing initial conditions are assumed. The problem is solved using the Boundary Element Method in the Laplace transformed domain. Integral expressions of the three-dimensional dynamic fundamental solutions for displacements and tractions are utilized. We employ the displacement boundary integral equations which are regularized using the static part of the dynamic anisotropic traction fundamental solution. For the spatial discretization of the boundary integral equations mixed boundary elements are adopted. The geometry of the considered domain is approximated with quadrilateral quadratic eight-noded elements. On the boundary elements displacements and tractions are interpolated using linear and constant shape functions, respectively. Time-domain solutions are obtained using suitable scheme for numerical inverse Laplace transform. The boundary-element solutions for the illustrative problem of an anisotropic elastic half space subjected to a Heaviside-type load are provided.

An experimental study on the contact problem of spherical indentation on the anisotropic elastic half-space based on imaging

The contact problem of spherical indentation on the anisotropic elastic half-space is considered based on an experimental method in this work. The mechanical anisotropy is simulated by uniaxially stretching isotropic material made of silica gel. A special optical system is adopted to capture the contact images during indentation tests. The experimental results further prove the reasonability of the assumption that the contact area of spherical indentation on the anisotropic half-space is elliptical. And the results also show that the direction of strengthening by the uniaxial tension is consistent with the minor axis of the elliptical contact area. Such results validate the associated theory. Besides, the error analysis is made to examine the reliability of the fitting results, where the relative errors are less than 0.8% under the influence of given drawbacks. The indentation modulus can be determined through experimental data. Results show that the indentation modulus monotonically increases with the increase of the prestrain value. In the case of isotropy, the initial elastic modulus is determined under the assumption of incompressibility. The result by indentation test is in a good agreement with that by uniaxial tensile test.

An approximate solution of the spherical indentation on a generally anisotropic elastic half-space

The indentation of a generally anisotropic elastic half-space by a rigid spherical indentor is considered. We derive an approximate equation for the eccentricity of the ellipsoidal contact region and an approximate expression for the indentation modulus. Our approximate formulae are first validated by ABAQUS-based numerical simulations, and then specialized to the particular case of a fibre-reinforced composite half-space. Numerical results are presented to show the dependence of the eccentricity and indentation modulus on the compressibility and fibre strength of the composite material. Possible application of our approximate formulae to the determination of elastic constants of fibre-reinforced materials is discussed.

Dynamic response of circular cavity and crack in anisotropic elastic half-space by out-plane waves

Due to the fact that anisotropic medium model is much common in earthquake engineering, dynamic response of a circular cavity and a crack in anisotropically elastic half-space under incident out-plane waves is examined in present work by using the methods of complex variable and Green's function. Based on the theory of Green's function, the method of complex variable is firstly developed to derive the corresponding Green's functions of half-space containing a circular cavity applied by an out-plane source load. Combined with “Crack-division”, a series of forces (the same in magnitude but opposite in direction to the stresses produced by incident SH-waves) are applied along the region where the crack will be constructed. Consequently, the total stresses will be zero, i.e., a crack can be created. Then, the displacement field while the crack coexists can be further derived. Finally, a mass of numerical examples are given to investigate the influence of different parameters on the dynamic stress concentration factors (DSCF) around the cavity and dynamic stress intensity factors (DSIF) at the crack tip.

Green’s functions for an anisotropic half-space and bimaterial incorporating anisotropic surface elasticity and surface van der Waals forces

In this paper we derive explicit expressions for the Green’s functions in the case of an anisotropic elastic half-space and bimaterial subjected to a line force and a line dislocation. In contrast to previous studies in this area, our analysis includes the contributions of both anisotropic surface elasticity and surface van der Waals interaction forces. By means of the Stroh sextic formalism, analytical continuation and the state-space approach, the corresponding boundary value problem is reduced to a system of six (for a half-space) or 12 (for a bimaterial) coupled first-order differential equations. By employing the orthogonality relations among the corresponding eigenvectors, the coupled system of differential equations is further decoupled to six (for a half-space) or 12 (for a bimaterial) independent first-order differential equations. The latter is solved analytically using exponential integrals. In addition, we identify four and seven non-zero intrinsic material lengths for a half-space and a bimaterial, respectively, due entirely to the incorporation of the surface elasticity and surface van der Waal forces. We prove that these material lengths can be only either real and positive or complex conjugates with positive real parts.

Displacement and stress fields due to finite faults and opening-mode fractures in an anisotropic elastic half-space

Displacement and stress fields due to finite faults and opening-mode fractures in an anisotropic elastic half-space

Explicit expression of polarization vector for surface waves, slip waves, Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space x2≥0 does not depend explicitly on the material property in the half-space x2≤0. It depends on the material property in the half-space x2≤0 implicitly through the interfacial slip wave speed υ. The same is true for the polarization vector for the material in the half-space x2≤0.

The Stroh formalism for elastic surface waves of general profile

The Stroh formalism is widely used in the study of surface waves on anisotropic elastic half-spaces, to analyse existence and for calculating the resultant wave speed. Normally, the formalism treats complex exponential solutions. However, since waves are non-dispersive, a generalization to waves having general waveform exists and is here found by various techniques. Fourier superposition yields a description in which displacements are expressed in terms of three copies of a single pair of conjugate harmonic functions. An equivalent representation involving just one analytic function also is deduced. Both these show that at the traction-free boundary just one component (typically the normal component) of displacement may be specified arbitrarily, the others then being specific combinations of it and its Hilbert transform. The algebra is closely related to that used for complex exponential waves, although the surface impedance matrix is replaced by a transfer matrix, which better embodies the scale invariance properties of the waves. Using the scale-invariance property of the boundary-value problem, a further derivation is presented in terms of real quantities.

Contact analysis for anisotropic elastic materials considering surface stress and surface elasticity

Contact analysis for an anisotropic elastic half-space is performed using the surface Green's function considering surface stress and surface elasticity. The discrete convolution and fast Fourier transform (DC–FFT) method and the conjugate gradient (CG) method was used to conduct the contact analysis. The influence coefficient in the contact analysis is calculated using the surface Green's function. The depth used employed in the surface Green's function considering surface stress and surface elasticity was decided such that the results of contact analysis of a rigid spherical indenter using Green's function for several values of depth agreed with the results of MD analysis. Several contact analyses were conducted in order to investigate the effects of material anisotropy and surface properties on the penetration depth for the ridge orientation in elliptic, Knoop, Vickers, and Berkovich indenters. When used to penetrate the (100) and (111) surfaces of Cu or Al, the penetration depth on indenter orientation clearly indicates material anisotropy in the elliptic and Knoop indenters, and indicates only a slight effect of material anisotropy in the Vickers and Berkovich indenters. The displacement field considering surface stress and surface elasticity is rounded in comparison with the shape of each crystal structure.

Surface waves guided by topography in an anisotropic elastic half-space

We consider the propagation of free surface waves on an elastic half-space that has a localized geometric inhomogeneity perpendicular to the direction of wave propagation (such waves are known as topography-guided surface waves). Our aim is to investigate how such a weak inhomogeneity modifies the surface-wave speed slightly. We first recover previously known results for isotropic materials and then present additional results for a generally anisotropic elastic half-space assuming only one plane of material symmetry. It is shown that a topography-guided surface wave in the present context may or may not propagate depending on a number of factors. In particular, they cannot propagate if the original two-dimensional surface wave on a flat half-space is supersonic with respect to the speed of anti-plane shear waves. For the case when a topography-guided surface wave may exist, the existence and computation of wave speed correction is reduced to the solution of a simple eigenvalue problem whose properties are previously well understood. As a by-product of our analysis, we deduce that there exists at least one topography-guided surface wave on an isotropic elastic half-space, and that it is unique when the geometric inhomogeneity has sufficiently small amplitude.

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.

Contact analyses for anisotropic half-space coated with an anisotropic layer: Effect of the anisotropy on the pressure distribution and contact area

A contact model using semi analytical methods, relying on elementary analytical solutions, has been developed. It is based on numerical techniques adapted to contact mechanics, with strong potential for inelastic, inhomogeneous or anisotropic materials. Recent developments aim to quantify displacements and stresses of a layered anisotropic elastic half space which is in contact with a rigid sphere. The influence of material properties and layer thickness on the contact problem solution will be more specifically analyzed.

Micro-mechanics models for an imperfect interface under anti-plane shear load: Hypersingular integral formulations

Abstract Two micro-mechanics models are proposed for an imperfect (weak) interface between two anisotropic elastic half-spaces under anti-plane shear load. The imperfect interface in the first model contains an array of periodically distributed micro-cracks with random lengths and positions. The second model, a simplified version of the first, is a three-phase model in which a period length of the interface contains a single representative micro-crack, perfectly bonded parts and an effective region. Both models are formulated in terms of hypersingular integral equations which may be solved by boundary element procedures to calculate the effective interface stiffness and the critical failure load of the interface.

Elastic Displacement and Stress Fields Induced by a Dislocation of Polygonal Shape in an Anisotropic Elastic Half-Space

The elastic displacement and stress fields due to a polygonal dislocation within an anisotropic homogeneous half-space are studied in this paper. Simple line integrals from 0 to π for the elastic fields are derived by applying the point-force Green’s functions in the corresponding half-space. Notably, the geometry of the polygonal dislocation is included entirely in the integrand easing integration for any arbitrarily shaped dislocation. We apply the proposed method to a hexagonal shaped dislocation loop with Burgers vector along [1¯ 1 0] lying on the crystallographic (1 1 1) slip plane within a half-space of a copper crystal. It is demonstrated numerically that the displacement jump condition on the dislocation loop surface and the traction-free condition on the surface of the half-space are both satisfied. On the free surface of the half-space, it is shown that the distributions of the hydrostatic stress (σ11 + σ22)/2 and pseudohydrostatic displacement (u1 + u2)/2 are both anti-symmetric, while the biaxial stress (σ11 − σ22)/2 and pseudobiaxial displacement (u1 − u2)/2 are both symmetric.

G010013 摩擦を考慮した異方性材料の接触解析([G01001]計算力学部門一般セッション(1):固体力学の基礎的な数値解析)

In this paper, the indenter is penetrated into an anisotropic elastic half-space with a pattern, and then a tangent force is applied to the indenter and judge a pressure distribution region and the relations of the shearing force. The partial slip contact problem was solved by a numerical procedure based on the conjugate gradient (CG) method and fast Fourier transform (DC-FFT) method. It is shown that the High shear stress and pressure occurrred at a corner of the surface pattern.