Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a linear elastic transversely isotropic bi-material full-space is presented. Three-dimensional point-load Green’s functions for stresses and displacements are given in complex-plane line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, within the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration. As illustrations, the present Green’s functions are analytically degenerated to the special cases such as half-space, surface and full-space Green’s functions. Some typical numerical examples are also given to show the general features of the bi-material Green’s functions.     

Thermopiezoelectric interaction of macro- and micro-cracks in piezoelectric medium

Considered is the interaction of macro-and micro-cracks in an anisotropic piezoelectric solid. The Green’s function and principle of superposition are used to formulate a system of singular integral equations for solving the unknown temperature discontinuity and elastic displacement-electric potential. The residual heat flux, stress and electric displacement on the microcrack are evaluated directly from the near-tip field of main crack. Numerical results for stress and electric displacement intensity factors in a three-crack system are obtained to illustrate the application of the method.     

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.     

Green’s function for T-stress of semi-infinite plane crack

Green’s function for the T-stress near a crack tip is addressed with an analytic function method for a semi-infinite crack lying in an elastical, isotropic, and infinite plate. The cracked plate is loaded by a single inclined concentrated force at an interior point. The complex potentials are obtained based on a superposition principle, which provide the solutions to the plane problems of elasticity. The regular parts of the potentials are extracted in an asymptotic analysis. Based on the regular parts, Gre...     

Analysis of quantum-dot-induced strain and electric fields in piezoelectric semiconductors of general anisotropy

Characteristics of the self-organized quantum dots (QDs) such as electron and hole energy levels and wave functions are dependent to the state of strain and electric field produced during the growing process of QDs in a semiconductor substrate. The calculation of the strain and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full three-dimensional and usually intricate shapes of the QDs. Numerical simulations are often performed by finite difference, finite element, or atomistic techniques, all require substantial computational time and memory. In this paper, we present a new Green’s function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity. Following the literature of micromechanics, the problem is formulated as an Eshelby inclusion problem of which the solution can be expressed by a volume-integral equation that involves the Green’s functions and the equivalent body-force of eiegenstrain. The volume integral is subsequently reduced to a line integral based on exploiting a unique structure of the Green’s functions. The final equations are cast in a form that most of the computational results can be repeatedly used for QDs at different locations—a very attractive feature for simulating large systems of QD arrays. The proposed algorithm has been implemented and validated by comparison with analytical solutions. Numerical simulations are presented for pyramidal QDs in the substrates of gallium arsenide (GaAs) (0 0 1).     

Microstructure damage related to stress–strain curve for grain composites

Mathematical model of micro-heterogeneous medium with random deformation and strength properties of microstructure is developed assuming that the tensor of macroscopic deformations is known for the structure. Green–Somigliana tensor is used to obtain the formulas for random stress distribution in microstructure elements. The probability of the stress exceeding the ultimate strength in an element determines the probability of fracture in this element and the relative micro-damage. The correlation functions of stochastic microstructure ultimate strength condition are calculated for various types of stress. Normal distribution is used to calculate the damage. The distribution density can be adjusted through the stress moments to the fourth order.Micro-fractions change the composite’s macro modules of elasticity. Therefore, changes the relationship between stress and strain. Setting an increment step on the macro-strain axis, the stress–strain curve is plotted taking into account changes in composite properties. Stress–strain curves are obtained for different types of load.The increase of the factor of safety corresponds with the reduction of microstructure damage permitted in the design. Critical microstructure damage also depends on the dispersion of the microstructure properties. It is shown that the microstructure properties of composite significantly influence the behavior of materials under load and the shape of stress–strain curve. Findings are compared with experiment data.     

Green’s functions for multi-phase isotropic laminated plates

This work develops a series of Green’s functions for multi-phase Kirchhoff isotropic laminated plates. First, we derive the Green’s functions for a composite laminated plate composed of two bonded dissimilar isotropic laminated semi-infinite plates. Second, the obtained results for bimaterials are judiciously applied to obtain the Green’s function solution for a circular elastic inclusion embedded in an infinite isotropic laminated plate. Third, Green’s functions for a composite space composed of an arbitrary number of wedges of different isotropic laminated plates are derived. Finally, we derive Green’s functions for a laminated plate with an elliptical and a parabolic boundary, respectively.     

Dynamic penny-shaped cracks in multilayer sandwich composites

A point force method is proposed for obtaining the dynamic elastic response of a multilayer sandwich composite in the presence of a penny-shaped crack under a harmonic loading. The sandwich composite is a multilayered solid whose lower half is the mirror image of the upper half with the center plane as the mirror. The crack is lying on the mirror plane of the composite. The solution of the mode I dynamic crack problem is formulated by integrating the Green’s function of a time-harmonic surface normal point force over the crack surface with an unknown point force distribution. The dual integral equations of the unknown point force distribution are established by considering the boundary conditions, which can be reduced to a Fredholm integral equation of the second kind. A complete solution of the crack problem under consideration can be obtained by solving this Fredholm integral equation. It will be shown that the results obtained by this approach are the same as some existing solutions.     

Green’s functions for plane problem under various boundary conditions and applications

Green’s functions of a point dislocation as well as a concentrated force for the plane problem of an infinite plane containing an arbitrarily shaped hole under stress, displacement, and mixed boundary conditions are stated. The Green’s functions are obtained in closed forms by using the complex stress function method along with the rational mapping function technique, which makes it possible to deal with relatively arbitrary configurations. The stress functions for these problems consist of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. These Green’s functions can be derived without carrying out any integration. The applications of the Green’s functions are demonstrated in studying the interaction of debonding and cracking from an inclusion with a line crack in an infinite plane subjected to remote uniform tension. The Green’s functions should have many other potential applications such as in boundary element method analysis. The boundary integral equations can be simplified by using the Green’s functions as the kernels.     

Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H, M, and N. With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P_i (i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.     

Antiplane time-harmonic green’s functions for a circular inhomogeneity with an imperfect interface

Two-dimensional antiplane time-harmonic Green’s functions for a circular inhomogeneity with an imperfect interface are derived. Here the linear spring model with vanishing thickness is employed to characterize the imperfect interface. Explicit expressions for the displacement and the stress fields induced by time-harmonic antiplane line forces located both in the unbounded matrix and in the circular inhomogeneity are presented. When the circular frequency approaches zero, our results reduce to those for the static case. Numerical results are presented to show the influence of the frequency and the imperfection of the interface on the stress and displacement fields.     

Two non-elliptical inhomogeneities with internal uniform stresses interacting with a mode-III crack

We use the method of Green's functions to analyze an inverse problem in which we aim to identify the shapes of two non-elliptical elastic inhomogeneities, embedded in an infinite matrix subjected to uniform remote stress, which enclose uniform stress distributions despite their interaction with a finite mode-III crack. The problem is reduced to an equivalent Cauchy singular integral equation, which is solved numerically using the Gauss–Chebyshev integration formula. The shapes of the two inhomogeneities and the corresponding location of the crack can then be determined by identifying a conformal mapping composed in part of a real density function obtained from the solution of the aforementioned singular integral equation. Several examples are given to demonstrate the solution.     

Crack analysis in semi-infinite transversely isotropic piezoelectric solid. I. Green’s functions

Three-dimensional fundamental solutions corresponding to a unit point force and point electric charge are obtained for a semi-infinite transversely isotropic piezoelectric solid. The free boundary is parallel to the plane of isotropy. They can be used as the Green’s function for solving the problem of a flat circular crack near the free surface which will be dealt with in Part II of this work.     

Rapid extension of a penny-shaped crack under torsion

The rapid extension of a penny-shaped crack under torsion is investigated. Both dynamic and quasi-static loading is considered. The wave motion is analyzed through a Green's function technique which leads to an integral equation for the stress field around the crack. Asymptotic expansions for the stress intensity and displacement rate intensity functions which are valid for a small time are obtained for the two types of loading. The propagation of the crack is analyzed through the balance of rates of energy criterion.     

Closed-form solution for a mode-III interfacial edge crack between two bonded dissimilar elastic strips

A closed-form solution is obtained for the problem of a mode-III interfacial edge crack between two bonded semi-infinite dissimilar elastic strips. A general out-of-plane displacement potential for the crack interacting with a screw dislocation or a line force is constructed using conformal mapping technique and existing dislocation solutions. Based on this displacement potential, the stress intensity factor (SIF, K_III) and the energy release rate (ERR, G_III) for the interfacial edge crack are obtained explicitly. It is shown that, in the limiting special cases, the obtained results coincide with the results available in the literature. The present solution can be used as the Green’s function to analyze interfacial edge cracks subjected to arbitrary anti-plane loadings. As an example, a formula is derived correcting the beam theory used in evaluation of SIF (K_III) and ERR (G_III) of bimaterials in the double cantilever beam (DCB) test configuration.     

含有多个涂层压电纤维复合材料的平面问题研究

压电纤维在未来的复合材料结构健康监测中具有重要作用．本文基于横观各向同性压电材料位移和应力连续条件以及经典的复势函数理论,讨论了同时受到平面内机械载荷和出平面电载荷作用时含有多个带涂层压电纤维的无限大线弹性基体的平面力学问题．首先将线弹性基体、涂层和压电纤维的应力场、位移场表示成复势函数,然后通过横观各向同性压电材料和线弹性材料的位移和应力连续条件确定复势函数表达式．将得到的复势函数表达式代入线弹性基体、涂层和压电纤维的的应力场、位移场公式可确定其应力场和位移场．最后,通过定量的案例讨论了涂层的材料属性对线弹性基体应力场的影响．案例分析表明涂层的材料属性对压电复合材料的应力场有重要的影响．     

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

Based on the extended Stroh formalism, we first derive the extended Green’s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi-material. These include Green’s functions of the extended dislocation, displacement discontinuities within a finite interval and the concentrated displacement discontinuities, all on the interface. The Green’s functions are then applied to obtain the integro-differential equation governing the interfacial crack. To eliminate the oscillating singularities associated with the delta function in the Green’s functions, we represent the delta function in terms of the Gaussian distribution function. In so doing, the integro-differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi-material with the extended displacement discontinuities being the unknowns. A simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities, along with the asymptotic expressions of the extended intensity factors and J-integral in terms of the discontinuities near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, and the influence of different extended loadings on the interfacial crack behaviors is further investigated.     

含椭圆孔非对称层合板的Green函数

运用作者建立的复合材料层合板问题的求解方法，得到了含椭圆孔和裂纹的无限大非对称层合板在广义集中载荷作用下的Ｇｒｅｅｎ函数．利用这些结果可研究含孔或裂纹的非对称层合板在集中载荷作用下的力学行为，还可结合边界元方法对复杂层合板结构建立有效的数值分析方法     

Subinterface crack in an anisotropic piezoelectric bimaterial

In this paper, the problem of a subinterface crack in an anisotropic piezoelectric bimaterial is analyzed. A system of singular integral equations is formulated for general anisotropic piezoelectric bimaterial with kernel functions expressed in complex form. For commonly used transversely isotropic piezoelectric materials, the kernel functions are given in real forms. By considering special properties of one of the bimaterial, various real kernel functions for half-plane problems with mechanical traction-free or displacement-fixed boundary conditions combined with different electric boundary conditions are obtained. Investigations of half-plane piezoelectric solids show that, particularly for the mechanical traction-free problem, the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary condition is applied on the boundary. However, for the piezoelectric bimaterial problem, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors. Instead, both elastic and electric properties of the bimaterial’s constants should be simultaneously taken into account for better accuracy of the generalized stress intensity factors.     

A general solution of 3-D quasi-steady-state problem of a moving heat source on a semi-infinite solid

The well-known Jaeger–Rosenthal asymptotic particular solution for the quasi-steady-state problem of moving heat source is proven to be inconsistent with the source constant intensity, especially at dimensionless trailing edge coordinates vx/a < −2. The problem is reduced to an equivalent Poisson’s equation by exponential transformation of moving coordinate scale. Using the method of images, the fundamental solution is found; the temperature rise function exponentially approximates to 0 along negative semi-axis. The temperature field in a semi-infinite solid for the general case of surface power intensity distribution is expressed, using the found Green’s function. The cases of point, line, and circular heat sources are considered. The found fundamental solution and particular solution for moving circular heat source explain the phenomena of martensite transformation in low-carbon steel substrate at relatively low source velocity 1.7 cm/s.