Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.     

Hertzian Contact of Anisotropic Piezoelectric Bodies

The Fourier transform method is applied to the Hertzian contact problem for anisotropic piezoelectric bodies. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. By presupposing the forms of the pressure and electric displacement distribution over the contact area, the problem is solved successfully; then the generalized displacements, stresses and strains are expressed by contour integrals. Details are presented in the case of special orthotropic piezoelectricity whose material constants satisfy six relations, which can be easily degenerated to the case of transverse isotropic piezoelectricity. It can be shown that the result gained in this paper is of a universal and compact form for a general material.Supported by the National Natural Science Foundation of China (No.10372003).     

BEM for modal analysis of 3-D anisotropic structures

The boundary element method for the modal anlysis of free vibration for 3-D anisotropic structures using particular solutions has been developed. The complete polynomials of order two are used to construct the particular solutions for general anisotropic materials. The numerical results for 3-D free vibration analysis of an isotropic cantilever beam by the method presented is in good agreement with the results using the Ritz technique. For anisotropic materials, the numerical results calculated from the proposed method are in good agreement with the results from MSC.NASTRAN. Project supported by the National Natural Science Foundation of China (No. 19872011).     

Superposing scheme for the three-dimensional Green’s functions of an anisotropic half-space

This paper presents a method of superposition for the half-space Green’s functions of a generally anisotropic material subjected to an interior point loading. The mathematical concept is based on the addition of a complementary term to the Green’s function in an anisotropic infinite domain. With the two-dimensional Fourier transformation, the complementary term is derived by solving the generalized Stroh eigenrelation and satisfying the boundary conditions on the free surface with the use of Green’s functions in the full-space case. The inverse Fourier transform leads to the contour integrals, which can be evaluated with the application of Cauchy residue theorem. Application of the present results is made to obtain analytical expression for the orthotropic materials which were not reported previously. The closed-form solutions for the transversely isotropic and isotropic materials derived directly from the solutions as being a special case are also given in this paper.     

Effect of dentin tubules on the mechanical properties of dentin Part III: Numerical analysis

Imitating a real tooth and the periodontal supporting tissues, we have established a 2D finite element model and carried out a numerical analysis based on the inhomogeneous and anisotropic (IA) stress-strain relation and strength model of dentin proposed in the preceding Parts I and II, and the conventional homogeneous and isotropic (HI) model, respectively. Quite a few cases of loadings for a non-defected and a defected tooth are considered. The numerical results show that the stress level predicted by the IA model is remarkably higher than that by the HI model, revealing that the effect of the dentin tubules should be taken into a serious consideration from the viewpoint of biomechanics. The project supported by the National Natural Science Foundation of China (19525207, 19891180) and the Tsinghua University Fundamental Research Foundation (Jc1999033)     

Stress-strain state of an anisotropic plate with curved cracks and thin rigid inclusions

We solve the bending problem for an anisotropic plate with flaws like smooth curved nonoverlapping through cracks and rigid inclusions. The problem is solved by the method of Lekhnitskii complex potentials specified as Cauchy type integrals over the flaw contours with an unknown integrand density function. We use the Sokhotskii—Plemelj formulas to reduce the boundary-value problem to a system of singular integral equations with the additional conditions that the displacements in the plate are single-valued when going around the cut contours and the equilibrium conditions for stress-free rigid inclusions. After the singular integrals are approximated by the Gauss-Chebyshev quadrature formulas, the problem is reduced to solving a system of linear algebraic equations. We study the local stress distribution near flaw tips. We analyze the mutual influence of flaws on the stress distribution character near their vertices and compare the well-known solutions for isotropic plates with the solutions obtained by passing to the limit in the anisotropy parameters (“weakly anisotropic material”) and by using the method proposed here.     

Simulation of anisotropic diffusion processes in fluids with smoothed particle hydrodynamics

Anisotropic diffusion phenomenon in fluids is simulated using smoothed particle hydrodynamics (SPH). A new SPH approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. Basic idea of the derivation is that anisotropic diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic. The coordinate transformation is a combination of a coordinate rotation and a scaling in accordance with diffusion tensor. Then, inverse coordinate transformation and particle discretization are applied to the integral to achieve ASPHAD. Noting that weight function used in the integral approximation has anisotropic smoothing length, which becomes isotropic under the inverse transformation. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusing coefficients. ASPHAD was numerically examined in some cases of isotropic and anisotropic diffusions of a contaminant in fluid, and the simulation results are very consistent with corresponding analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

The explicit formulations for theL-integral

In this paper, a simple but inberent relation between theL-integral and the Bueckner work conjugate integral is proved for crack problems in isotropic, anisotropic, and dissimilar materials, respectively. It is found that, in the above-mentioned three cases, theL-integral, from the mathematical point of view as well as in principle, arises from Betti's reciprocal theorem. This means that the Bueckner work conjugate integral is a more general path-independent integral than the others since any other path-independent integrals could be derived by using the Bueckner integral while choosing a different subsidiary stress-displacement field.     

Elastodynamic analysis of inhomogeneous anisotropic bodies

The integral equation method is presented for elastodynamic problems of inhomogeneous anisotropic bodies. Since fundamental solutions are not available for general inhomogeneous anisotropic media, we employ the fundamental solution for homogeneous elastostatics. The terms induced by material inhomogeneity and inertia force are regarded as body forces in elastostatics, and evaluated in the form of volume integrals. The scattering problems of elastic waves by inhomogeneous anisotropic inclusions are investigated for some test cases. Numerical results show the significant effects of inhomogeneity and anisotropy of materials on wave propagations.     

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In this paper we study the antiplane problem of concentrated point force moving with constant velocity and oscillating with constant frequency in unbounded homogeneous anisotropic elastic medium.The explicit representation of the elastodynamic Green's function is obtained by using Fourier integral transform techniques for all rates of source motion as a sum of the integrals over the finite interval. The dynamic and quasistatic components of the Green's function are extracted. The stationary phase method is applied to derive an asymptotic approximation at the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too.It is shown that the motion brings some differences in the far field properties, such as, for example, fast and slow waves appearance under superseismic motion and modification of the wave propagation zones and their numbers.The case of isotropic medium is considered separately. For isotropic material all main formulae are obtained in explicit forms.     

两种各向异性材料界面共线裂纹的反平面问题

本文研究两种各向异性材料界面共线裂纹的反平面剪切问题。利用复变函数方法，提出了一般问题公式和某些实际重要问题的封闭形式解。考察了裂纹尖端附近的应力分布并给出了应力强度因子公式。从本文解签的特殊情形，可以直接导出两种各向同性材料界面裂纹，均匀各向异性材料共线裂纹以及均匀各向同性材料共线裂纹的相应问题公式，其中包括已有的经典结果。     

NUMERICAL SIMULATION OF ELASTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

A new numerical simulation algorithm is presented for the elastic wave propagation in heterogeneous anisotropic media. We make discretization of the computational domain by using triangular and quadrangular grids. The scheme is based on integral equilibrium at each node to simulate the elastic wave propagation in heterogeneous anisotropic media. The scheme is very flexible in dealing with arbitrary surface topography, inner openings, liquid-solid boundaries and irregular interfaces. Moreover, the free-surface condition of complex geometrical boundaries can be satisfied naturally. This work is an extension of the grid method for the elastic wave propagation in heterogeneous isotropic media, and a quadrangular grid with low computational cost is also introduced. Project supported by the National Natural Science Foundation of China(19672016).     

A large strain plasticity model for anisotropic materials — composite material application

In this work a generalized anisotropic model in large strains based on the classical isotropic plasticity theory is presented. The anisotropic theory is based on the concept of mapped tensors from the anisotropic real space to the isotropic fictitious one. In classical orthotropy theories it is necessary to use a special constitutive law for each material. The proposed theory is a generalization of classical theories and allows the use of models and algorithms developed for isotropic materials. It is based on establishing a one-to-one relationship between the behavior of an anisotropic real material and that of an isotropic fictitious one. Therefore, the problem is solved in the isotropic fictious space and the results are transported to the real field. This theory is applied to simulate the behavior of each material in the composite. The whole behavior of the composite is modeled by incorporating the anisotropic model within a model based on a modified mixing theory.     

两种各向异性材料界面周期裂纹的反平面问题

研究两种各向异性材料界面含周期裂纹的反平面剪切问题，运用复变函数方法，获得了封闭形式解答，并给出了应力强度因子公式。从本文解签的特殊情形，可直接导出均匀各向异性材料共线裂纹，两种各向同性材料界面裂纹的相应问题公式。     

Explicit elasticity-conductivity connections for composites with anisotropic inhomogeneities

Explicit elasticity-conductivity connections for anisotropic heterogeneous materials were recently derived and verified experimentally by the present authors. The constituents were assumed isotropic, so that the overall anisotropy was due solely due to nonrandom orientations of inhomogeneities. Motivated by the materials science applications that deal with strongly anisotropic inclusions, we derive alternative elasticity-conductivity connections that cover these cases. They hold for a broad class of orientation distributions—up to three families of parallel inhomogeneities forming arbitrary angles with each other, with moderate orientation scatter allowed in each of the families. In the case of the isotropic inhomogeneities, this form has substantially better accuracy than the approximate connections derived earlier. The results are compared with experimental data on fiber-reinforced plastics. The agreement is quite good for the entire set of the effective anisotropic constants, and it is achieved without any fitting parameters.     

The degeneration of image singularities from anisotropic materials to isotropic materials for an elastic half-plane

The degeneration of image singularities from an anisotropic material to an isotropic material for a half-plane is discussed in this study. The Green’s functions for anisotropic and isotropic half-planes with traction free boundary subjected to concentrated forces and dislocations have been obtained by many authors. It was commonly accepted that the solution of isotropic problem cannot be derived from anisotropic solutions. However, we believe that this possibility exists as we will demonstrate in this paper. Anisotropic materials include only image singularities of order O(1/r) (i.e., forces and dislocations) existing on image points. There are many image points for anisotropic materials and the locations of these image points depend on the material constants. However, isotropic materials have only one image point with higher order image singularities (O(1/r²), O(1/r³)). From the analysis provided in this study, it is found that the higher order image singularities for an isotropic half-plane are generated by combining the concentrated forces and dislocations when an anisotropic material degenerates to an isotropic material. The solutions of higher order image singularities for isotropic material are dependent. Therefore, these image singularities can be combined to form only three or four simpler image singularities acting on an image point of the isotropic material.     

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.     

Stress in transversely isotropic half-space with typical loads acting on its surface

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