Green’s functions for transversely isotropic magnetoelectroelastic media

Based on the governing equations of transversely isotropic magnetoelectroelastic media, four general solutions on the cases of distinct eigenvalues and multiple eigenvalues are given and expressed in five mono-harmonic displacement functions. Then, based on these general solutions, employing the trial-and-error method, the three-dimensional Green’s functions of infinite, two-phase and semi-infinite magnetoelectroelastic media under point forces, point charge and magnetic monopole are all presented in terms of elementary functions for all cases of distinct eigenvalues and multiple eigenvalues. Numerical results are also presented.     

A combined nonequilibrium Green’s function/density-functional theory study of electrical conducting properties of artificial DNA duplexes

DNA duplexes have attracted much attention as a primary candidate for nanowires possessing self-organizing capability. To employ DNA duplexes as nanowires, however, a major drawback must be overcome; the guanine bases undergo oxidative degradation in a hole transport through DNA duplexes, which is likely caused by the presence of adjoining adenine bases that do not effectively mediate the charge transport through DNA duplexes. To overcome the drawback, several artificial nucleobases based on adenine have been designed and tested, confirming that the artificial nucleobase-containing DNA duplexes do not suffer from such an oxidative damage and exhibit high efficiency in hole transport through the DNA duplexes. In the present study, we examine the electrical conducting properties of these artificial DNA duplexes by use of nonequilibrium Green’s function and density-functional theory methods. The results explicate the origin of the experimentally observed high conductivity through the DNA duplexes containing the artificial DNA bases. We also put forth a computer-aided design of novel artificial DNA bases with low ionization energies, and examine the electrical conductivity of the DNA duplexes containing the designer nucleobases for potential use as highly conductive nanowires.     

Effect of hydration on electrical conductivity of DNA duplex: Green’s function study combined with DFT

The electrical conductivity of DNA duplex is affected by many factors such as DNA base sequence, hydration of DNA duplex, connecting configuration between DNA duplex and electrodes, thermal fluctuation of DNA duplex structure. We here investigate the electrical conducting properties of DNA duplexes sandwiched between Au electrodes by molecular simulations using nonequilibrium Green’s function method coupled with density functional theory. The results reveal the dependence of electrical conductivity on DNA base sequence as well as hydration, which are in qualitative agreement with experiment. The present results indicate the important role played by hydrating water molecules in the electrical conductivity through DNA duplexes.     

A note on the divergence problem arising in calculation of Green’s function for a 2D piezoelectric thin film strip containing straight line defects and free boundaries: Fourier approach for bodies of unrestricted anisotropy

In this paper we derive a set of novel formulas for computation of the Green’s function and the coupled electro-elastic fields in a 2D piezoelectric strip with free boundaries and containing a distribution of straight line defects. The strip is assumed to be of unrestricted anisotropy, but allowing piezoelectricity, and in this sense situation is more general than in the available literature where only cubic symmetry was investigated. We employ a set of already known analytic formulas for the Fourier amplitude of the Green’s function and the corresponding electro-elastic fields. The key novelty of this paper is solution for the divergence problem occurring during integration of the Fourier amplitude. This problem is caused by poles at k = 0 in various matrix components of the amplitude. From purely mathematical point of view such poles lead to quantities which do not tend to zero at infinity, and this situation is clearly unphysical. To resolve this issue it is demonstrated by means of rigorous analysis that when some additional physical conditions are imposed, physical fields exhibit regular behavior at infinity - the poles do not contribute. Nevertheless, they lead to irremovable numerical ∞ − ∞ uncertainties spreading over the whole domain of integration. This motivates us to compute exact formulas for all these poles to enable engineering calculations involving the system in question.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

A note on the divergence problem arising in calculation of Green’s function for a 2D piezoelectric thin film strip with free boundaries and containing straight line defects: Fourier approach for bodies of cubic symmetry

Analytic formulas for the Green’s function and the coupled electro-elastic fields for a 2D piezoelectric strip with free boundaries and containing a distribution of straight line defects have already been found some years ago. These formulas exploit the well-known Stroh formalism and the Fourier approach, so the result is given as the Fourier integral and therefore its numerical implementation should pose no problem. However, in this note we show that for the case of cubic symmetry this form of the Green’s function contains strong divergences, excluding possibilities of direct application of well-known numerical schemes. It is also shown that these divergences translate to divergences of the corresponding electro-elastic fields of a single defect. By means of a rigorous analysis it is demonstrated that imposing physical conditions implied by the nature of the problem all of these divergences cancel and the final, physical result exhibits expected, regular behavior at infinity. Unfortunately, although the nature of this problem is purely mathematical, it leads to irremovable numerical ∞ − ∞ uncertainties which tend to spread over the whole Fourier domain and severely impede engineering applications of the Green’s function. This motivates us to compute the exact form of all divergent terms. These novel formulas will serve as a guide when establishing numerically stable algorithms for engineering computations involving the system in question.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Potential discontinuity simulations with the numerical Green’s function in steady and transient problems

This paper applies the numerical Green’s function (NGF) boundary element formulation (BEM) first in standard form to solve the Laplace equation and then, coupled to the operational quadrature method (OQM), to solve time domain problems (TD-BEM). Both involve the analysis of potential discontinuities in the respective scalar model simulation. The implementation of the associated Green’s function acting as the fundamental solution is advantageous since element discretization of actual discontinuity surfaces are no longer required. In the OQM the convolution integral is substituted by a quadrature formula, whose weights are computed using the fundamental solution in the Laplace domain, producing the direct solution to the problem in the time domain. Applications of the NGF to problems involving the Laplace equation and its transient counterpart are presented for two-dimensional potential flow examples, confirming that the formulation is stable and accurate.     

The effect of second principal stress on the fatigue propagation of mode I surface crack in Al2O3/Al alloy composites

Using a plate made of A2017-T6 metal matrix composites reinforced with 10 volume % and 20 volume % Al₂O₃ particles and Al alloy possesses the same composition as matrix alloy, the crack propagation rate da/dN of a mode I surface crack by the simultaneous action of plane bending and cyclic torsion are studied. And the effects of crack tip opening stress σ_top, crack opening displacement COD, biaxial stress ratio C (=second principal stress/first principal stress) and the surface roughness of crack section are examined. When stress intensity factor range ΔK is lower than the specific level, da/dN decreases with the increase of volume fraction of Al₂O₃ in C=0 and C=−0.55. But, da/dN of Al alloy becomes minimum in C=−1 and the effect of Al₂O₃ particles disappears. σ_top rises with the increase of volume fraction of Al₂O₃ particles and the decline of C. On the other hand, COD doesn’t always rise with the decline of C. These phenomena can be explained by the residual compressive stress formed at the surface layer of the specimen by the fatigue test and the surface roughness of crack section.     

On the Mode I stress intensity factor for an external circular crack with fibre bridging

This paper presents a theoretical analysis of an external matrix crack located in a unidirectional fibre-reinforced elastic solid modelled as a transversely isotropic material. The presence of matrix cracking with fibre continuity introduces bridging action that has an influence on the stress intensity factors at the crack tip of the external crack. This paper presents a model for the bridged crack, where the fibre ligaments induce a constant displacement-dependent traction constraint over the external crack. This gives rise to a Fredholm integral equation of the second kind, which can be solved in an approximate fashion. We examine the specific problem where the bridged external circular crack is loaded by a doublet of concentrated forces. Numerical results are presented to illustrate the influence of the fibre–matrix modular ratio and the location of the loading on the bridged-crack opening mode stress intensity factor.     

A direct method for calculating the stress intensity factor in BEM

The proposed algorithm employs singular crack tip elements in which the stress intensity factor appears as a degree of freedom. The additional degrees of freedom are compensated by constraint conditions which originate from imposing continuity across elements and a contour integration formula. The two benchmark problems indicate the proposed algorithm can accurately predict the stress intensity factor and the distribution of the primary and secondary variables in fracture problems.     

Weight function for an external circumferential semielliptical crack in a cylinder

In this paper, the weight function was extracted at the deepest point of a semielliptical circumferential crack. The crack is assumed to exist on the outer surface of the cylinder. For this purpose, the three‐dimensional finite element method was accomplished to specify two reference loads, which are indispensable for determining the weight function. The verification study confirms the accuracy of the derived weight function under prescribed mechanical loading on the crack surfaces. There is consistency among the solution results compared with those in the literature. The second part describes the application of the weight function for the thermal boundary conditions. Steady‐state thermal stress intensity factors are demonstrated using the weight function and presented as a closed‐form solution. The results were compared with the finite element data on the special case of thermal loading, and good agreement is obtained.     

Three-dimensional Green’s functions for a steady point heat source in a functionally graded half-space and some related problems

Three-dimensional Green’s functions are derived for a steady point heat source in a functionally graded half-space where the thermal conductivity varies exponentially along an arbitrary direction. We first introduce an auxiliary function which satisfies an inhomogeneous Helmholtz equation. Then by virtue of the image method which was first proposed by Sommerfeld for the homogeneous half-space Green’s function of a steady point heat source, we arrive at an explicit expression for this function. Finally with this auxiliary function, we derive the three-dimensional Green’s functions due to a steady point heat source in a functionally graded half-space. Also investigated in this paper are the temperature field induced by a point heat source moving at a constant speed in a functionally graded full-space; the electric potential due to a static point electric charge in a dielectric full-space with electric field gradient effects; and the two-dimensional time-harmonic dynamic Green’s function for homogeneous and functionally graded materials with strain gradient effects.     

Calculation of KII in crack face contacts using X-FEM. Application to fretting fatigue

In this work, the modeling of LEFM problems that imply crack face closure and contact using the extended finite element method (X-FEM) is presented aiming at its application to fretting fatigue problems. An assessment of the accuracy in the calculation of K_II is performed for two different techniques to model crack face contacts in X-FEM: one is based on the use of additional elements to establish the contact and the other on a segment-to-segment (or mortar) approach. It is concluded that only the segment-to-segment approach can lead to optimal convergence rates of the error in K_II. The crack face contact modeling has also been applied to a fretting fatigue problem, where the estimation of K_II under crack closure conditions plays an important role in the stage I of fatigue crack propagation. The effect of the crack face friction coefficient has been studied and its influence on the range of K_II has been ascertained during loading and unloading cycles.     

Finite element implementation of thermal weight function method for calculating transient SIFs of a body subjected to thermal shock

The thermal weight function (TWF) is a universal function, which is dependent only on the crack configuration and body geometry, and is independent of temperature fields. The TWF method is especially suitable for determining the variation of transient stress intensity factors (SIFs) of a cracked body subjected to thermal shock. TWF is independent of time during thermal shock, so the whole variation of transient SIFs can be directly calculated through integration of the products of TWF and transient temperatures and temperature gradients. The repeated determinations of the distributions of stresses (or displacements) fields for individual time instants are thus avoided in the TWF method, which are necessary when the direct method through analyses of thermo-elasticity or the mechanical weight function (MWF) method is applied. The finite element implementation of the TWF method for Mode I in plane stress, plane strain and axisymmetric problems are presented in this paper. In the TWF method, the integration should be carried out around the boundary as well as over the whole volume. So, it is a practical and useful way to develop an integrated system of programs for solving the thermal shock problems by means of the TWF method, which has been developed by authors. Examples show that the scheme shown in this paper is of very high efficiency and of good accuracy.     

弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的有限元分析

对表面裂纹复合型应力强度因子的研究一直是线弹性断裂力学中的重要课题,例如弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的计算,到现在也没有一个正确的分析解。考虑到裂尖的应力奇异性,在裂纹前沿手动设置三维奇异单元,用三维有限元法中的1/4点位移法计算弯扭组合载荷下圆管表面椭圆裂纹前沿的Ⅰ型、Ⅱ型和Ⅲ型应力强度因子,并分析其随裂纹深度增加时的变化规律。运用该方法计算了有关模型的应力强度因子,并与该模型的实验值进行了比较,计算结果和实验结果吻合良好。     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd     

On the static equilibrium of an elastic orthotropic medium with an arbitrarily oriented elliptical crack

We analyze the problem of the stress distribution in an elastic orthotropic medium with an arbitrarily oriented elliptical crack. To construct the problem solution, the Willis approach is used which is based on the triple Fourier transformation of spatial variables and Fourier-image of Green’s function for an infinite anisotropic space. The investigation results in special cases are compared with the data of other authors. The effect of the elliptical crack orientation in an orthotropic space on the distribution of the stress intensity factors along its contour is studied. __________ Translated from Problemy Prochnosti, No. 4, pp. 146–159, July–August, 2007.     

Internal–external circumferential crack behaviour in the cement layer of total hip replacement

This study aimed to investigate crack behaviour at the internal and external surfaces of the cement layer in total hip replacement. A three‐dimensional model of the femur with the cemented prosthesis was developed and analysed. Cracks were placed on the internal, external and both internal and external surfaces of the cement layer. Stress intensity factors were measured during gait. Results revealed that the stress intensity factors modes I and III were the most dominant in the crack propagation in the cement layer. The domain of mode I was the medial and lateral sides of the cement layer. Meanwhile, the domain of mode III was the anterior and posterior sides of the cement layer. The stress intensity factor and distance from the distal end indicated an inverse relationship. The internal and external cracks had no significant interaction. Moreover, stress intensity factors at the external surface of the cement layer were higher than those on the internal surface.     

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.