The seismic pulse in a semi-infinite medium

Summary In a foregoing paper the present author developed methods for studying the transient field from a vertical electric antenna placed in the vicinity of the plane boundary of two semi-infinite dielectric media.As the theory involved is applicable to the comparable elastodynamic pulse problem the present paper deals with the field from a buried transient longitudinal source in an elastic half space.The method appears to be relatively simple and is also applicable to the more general problem in which two elastic semi-infinite solids are separated by a plane boundary.     

Scattering of elastic waves by an inclusion with an interface crack

A method of perturbation is used to derive an integral representation of the displacement field for the scattering of a plane wave from an inclusion with an interface crack. In the long-wave approximation it is shown that the solution of only an associated static problem is required and formal expressions are derived for the scattered far field amplitudes and scattering cross section. In the case of a cylindrical inclusion the solution of the associated static problem is then used to find in a closed form the corresponding expressions for plane incident P- and S-waves.     

Temperature field at time of pulse current discharge in metal structure with elliptical embedding crack

Theoretical analysis is made on the temperature field at the time of pulse current discharge in a metal structure with an elliptical embedding crack. In finding the temperature field, analogy between the current flow through an elliptical embedding crack and the fluid flow through a barrier is made based on the similarity principle. Boundary conditions derived from this theory are introduced so that the distribution of current density and the temperature field expressions can be obtained. The study provides a theoretic basis to the applications of stopping spatial crack with electromagnetic heating.     

Gage-length errors in the resolution of dispersive stress waves

The effect of measuring the time history of a passing dispersive wave with a finite-length gage is to introduce time and amplitude errors between the actual values and the measured values of the wave function. In this paper, mathematical expressions relating the size of these errors to the gage length, properties of the material, and spectral content of the incident wave are derived. These expressions are used to predict the response of a long, resistance strain gage, attached in the longitudinal direction on the lateral surface of a long slender rod of a linear viscoelastic material (Lexan), to a short pulse. The shape of the pulse just before and just after the region of the long gage was measured by means of very short gages. The dispersion and attenuation properties of the material were obtained from the short-gage data. This information and the previously derived mathematical-correction equations were used to predict the response of the long gage. Good agreement was obtained between the predicted and measured values of the response of the long gage. In addition, the correction equations and the response of the long gage were used to predict the incident pulse. Good agreement with the known experimental values was obtained.     

The fundamental equations in finite element method of coupled thermo-elastic plane problem

The fundamental equations in finite element method for unsteady temperature field elastic plane problem are derived on the bases of variational principle of coupled thermoelastic problems. In these derivations, elastic plane is divided into three nodes triangular elements, and time interval is divided into linear time elements, in which all the variables, including displacements and temperatures at various nodal points, are varied linearly with time. Two coupled sets of linear algebraic equations of all the unknown displacements and temperatures at every nodal point in every instant (i.e. the terminal values of time elements) are obtained. They are the fundamental equations of the said problem.The total energy in elastic body not only contains the potential energy and heat energy but also contains the kinetic energy, if the rate of change of temperature field with respect to the time in thermoelastic problem is large enough. And the change of displacement is included in the equations of heat conduction. For this reason the variational principle of coupled thermoelastic problems is employed. [1] In this paper, expressions of this principle for plane problems are given. The discretization is carried on then, and Hamilton's action and the potential action of heat flow of elements are derived. Finally they are assembled, so as to get the polar value of the action. And thus the groups of linear algebraic equations in matrix form are obtained.     

On the screw dislocation in a functionally graded piezoelectric plane and half-plane

In this paper closed-form expressions of the electroelastic field induced by a piezoelectric screw dislocation in a functionally graded piezoelectric plane and half-plane are derived. The material properties are assumed to vary exponentially along the x and y-directions. The solution for a screw dislocation in a functionally graded piezoelectric plane is obtained through introduction of two generalized stress functions. The solution for a screw dislocation in a functionally graded piezoelectric half-plane is derived by using the method of image. It is also found that the interaction between a piezoelectric screw dislocation and a circular insulating hole in the functionally graded piezoelectric material can be solved by using series expansion method.     

Analytical solution of a flow field for weak Mach reflection over a plane surface

In this paper, we present an analytical solution for the flow field of single weak Mach reflection caused by an advancing plane shock wave over a simple wedge surface. We developed an improvement of Lighthill's linearized theory for the correction due to nonlinearity of the flow field through a singular perturbation. The expressions obtained, including the one for the triple point path, compared favorably with existing experimental, computational, and theoretical results. PACS 01.50.Kw; 47.15.Pn Communicated by K. Takayama     

Determination of stress levels for dynamic fracture of oil shale

The dynamic tensile-stress amplitudes necessary to cause complete spall in unconfined oil-shale samples were experimentally determined in the laboratory for pulse durations typically encountered in the field (60–100 μs). Tensile stresses were generated by free-surface reflection of compressive pulses, and the stress-time history at the free surface was obtained by numerically differentiating the displacement-time history monitored by a fiber-optic proximity sensor. A computer code was written to display the stress profile, obtained experimentally, over the length of the specimen as time increased. In this manner, the amplitude of the resultant stress wave (incident plus reflected) was determined at the measured spall plane. By using many specimens, the fracture thresholds for these pulse durations were determined. Spall thresholds for competent shale of 25-gal/ton yield were found to be approximately 1800 psi (12.4 MPa) for pulses propagating normal to the bedding plane, and nearly 5,000 psi (34.5 MPa) for pulses propagating parallel to the bedding plane. For shale of 34 gal/ton yield, the spall-threshold levels were found to be 1200 psi (8.3 MPa) and 2700 psi (18.6 MPa) for pulses propagating normal and parallel, respectively, to the bedding plane.     

Exact solution of a semi-infinite crack in an infinite piezoelectric body

Summary The paper presents an exact and complete solution of the problem of a semi-infinite plane crack in an infinite transversely isotropic piezoelectric body. The upper and lower crack faces are assumed to be loaded symmetrically by a couple of normal point forces in opposite directions and a couple of point charges. The solution is derived through a limiting procedure from the one of a penny-shaped crack. The expressions for the elastoelectric field are given in terms of elementary functions. Received 10 August 1998; accepted for publication 18 November 1998     

Visualising elastic anisotropy: theoretical background and computational implementation

In this article, we present the technical realisation for visualisations of characteristic parameters of the fourth-order elasticity tensor, which is classified by three-dimensional symmetry groups. Hereby, expressions for spatial representations of Young’s modulus and bulk modulus as well as plane representations of shear modulus and Poisson’s ratio are derived and transferred into a comprehensible form to computer algebra systems. Additionally, we present approaches for spatial representations of both latter parameters. These three- and two-dimensional representations are implemented into the software MATrix LABoratory. Exemplary representations of characteristic materials complete the present treatise.     

MAGNETO-THERMO-ELASTIC COUPLED DYNAMICS EQUATIONS OF AXIALLY MOVING CARRY CURRENT PLATE IN MAGNETIC FIELD

针对磁场环境中轴向运动载电流导电板磁热弹性耦合动力学建模问题进行研究. 考虑几何非线性和热效应条件下, 给出薄板运动的动能、应变能以及外力虚功的表达式.应用哈密顿变分原 理, 推得力、运动、电、磁和热效应相互作用下轴向运动导电板的非线性磁热弹性耦合振动方程.基于麦克斯韦电磁场方程, 考虑相应的电磁本构关系和电磁边界条件, 推得磁场环境中轴向运动载电流导电板满足的电动力学方程和所受电磁力表达式, 并给出焦耳热作用下耦合形式的热传导方程. 算例表明, 磁场等参量对动力学系统分岔特性有明显影响.所得结果可为此类问题的进一步求解和分 析提供理论参考.     

An elastic wave scattering problem relating to the rapid movement of a fracture in a plate: Part II — Numerical results

In Part I of this two part paper, we developed analytical expressions for the crack face displacements due to an applied point traction as well as a moving point source of dilatation. In this sequel to Part I, we present numerical results for the crack face displacements in the form of displacement-time records at fixed locations on the crack face. For the stationary point traction problem, the results are more or less in keeping with the predictions made in Part I, with the Rayleigh pulse giving a large contribution for an applied step load.In the case of the moving point microfracture with a step time dependence, in addition to the diffracted compressional, shear and Rayleigh arrivals there is observed the evolution of a new pulse whose amplitude and position on the time record shifts changes if the calculations are done for a different crack speed. Results are also presented for a microfracture with an oscillatory time dependence. Once again, the first arrival of the surface pulse appears to be the most prominent arrival on the record. Further explorations of the parameters and source functions would be required before comparisons with experimental data can be made.     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

Anisotropic plastic stress field at a mixed-mode crack tip under plane and anti-plane strain

On condition that any perfectly plastic stress component at a crack tip is nothingbut the function ofθ.by making use of equilibrium equations,anisotropic plastic stress-strain-rate relations,compatibility equations and Hill anisotropic plastic yieldcondition,in the present paper,we derive the generally analytical expressions of theanisotropic plastic stress field at a mixed-mode crack tip under plane and anti-planestrain.Applying these generally analytical expressions to the mixed-mode cracks,wecan obtain the analytical expressions of anisotropic plastic stress fields at the tips ofmixed-modeⅠ-Ⅲ,Ⅱ-ⅢandⅠ-Ⅱ-Ⅲcracks.     

On perfectly stress field at a mixed-mode crack tip under plane and anti-plane strain

In [1], under the condition that all the perfectly plastic stress components at a crack tip are functions of ϕ only, making use of equilibrium equations, stress-strain rate relations, compatibility equations and yield condition. Lin derived the general analytical expressions of the perfectly plastic stress field at a mixed-mode crack tip under plane and anti-plane strain. But in [1] there were several restrictions on the proportionality factor γ in the stress-strain rate relations, such as supposing that γ is independent of ϕ and supposing that γ=c or cr⁻¹. In this paper, we abolish these restrictions. The cases in [1], γ=cr^d (n=0 or-1) are the special cases of this paper.     

WAVELET METHOD FOR CALCULATING THE DEFECT STATES OF TWO-DIMENSIONAL PHONONIC CRYSTALS

Based on the variational theory, a wavelet-based numerical method is developed to calculate the defect states of acoustic waves in two-dimensional phononic crystals with point and line defects. The supercell technique is applied. By expanding the displacement field and the material constants （mass density and elastic stiffness） in periodic wavelets, the explicit formulations of an eigenvalue problem for the plane harmonic bulk waves in such a phononic structure are derived. The point and line defect states in solid-liquid and solid-solid systems are calculated. Comparisons of the present results with those measured experimentally or those from the plane wave expansion method show that the present method can yield accurate results with faster convergence and less computing time.     

Time domain radiation and scattering by thin wires

Direct time domain solutions for radiation from and scattering by thin conducting wires are considered. The problem is formulated in terms of two coupled integrodifferential equations derived from the retarded potentials, the continuity equation, and the boundary conditions for the wire. Solution of these equations is effected using the method of moments resulting in a set of simultaneous time iterative matrix equations. A time domain reciprocity theorem suitable for thin wire objects is also presented. The theorem demonstrates the relationship between reciprocity and the adjoint operator for the problem. Two moment solutions are presented for straight wires: (1) a point tested solution, and (2) a pulse tested solution. The radiation field is found by direct application of the reciprocity theorem. All solutions are presented as algorithms suitable for computation. The algorithms are time iterative by nature, and inexpensive in terms of computer time. Computed results are presented for the straight wire scatterer excited by a plane wave with unit step time dependence. The end fire effects predicted by traveling wave theory are observed in the scattered field results. The source of these effects is shown to be the end of the wire last intercepted by the incident field. Additional results are presented for the straight wire antenna excited by a unit step voltage applied at an arbitrary driving point. The computed driving point current is compared to the results derived for the case of an infinite wire antenna over the time interval that the comparison is valid.     

THE PROBLEM OF I-II COMBINED PLANE CRACK SOLVED WITH WESTERGAARD STRESS FUNCTION

直接获得I-II复合型平面裂纹问题裂纹尖端区域的应力场是一个比较复杂的问题,在此应用Westergaard应力函数求解I-II复合型平面裂纹问题,导出了裂纹尖端区域应力分量的表达式。该方法推导过程简单,物理概念清晰,其结果与一般断裂力学教材和文献中的结果一致。同时,应用叠加原理将裂纹面上的作用力转化为裂纹外边界的受力,给出了解决裂纹面上有作用力的I-II复合型平面裂纹问题的解题方法。     

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that those plane stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.