For a fairly large group of boundary value problems, especially when the problem domain is of infinite extend, the boundary element method (BEM) can be faster and more accurate than the finite element method (FEM) and finite difference method (FDM). However, in handling three dimensional elasticity problems of general anisotropy and piezoelectricity, BEM faces challenges. That is because the Green's functions are costly to compute and, as a result numerical integration of the Green's functions over a large number of boundary elements become computationally intensive. In this work, integration over the boundary elements is carried out analytically. In doing so, significant saving in computing time is achieved.
Following the framework of Wang and Achenback who used Radon transform, three dimensional Green's functions for solids of general anisotropy and piezoelectricity are derived in the form of a line integral over a half unit circle. Kernels in these line integrals are simple algebraic functions, which are similar in structure as the two-dimensional solutions of anisotropic elasticity. Taking advantage of the simple structure of the kernels, we (1) exchange integration orders between the surface integration over the boundary elements and the line integration over the half unit circle, (2) analytically carry out the surface integration over the boundary elements, and (3) leave the line integrals over the half unit circle to be computed numerically. The line integrals are non-singular and easy to compute numerically.
The BEM developed is the first 3D BEM for the general anisotropic and piezoelectric solids.
Windows and C++ based pre and post processors for the 3D static BEM for general anisotropic solids are developed to facilitate the input and output of the three dimensional analysis.
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.
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 exploiting a unique structure of the Green's functions. The final equations are cast in a form that most of the computational results are performed and stored for one QD so that they can be repeatedly used for QDs at different locations - a very attractive feature for simulating large systems of QD arrays.
This work presents a new Green's function approach for 3-D inclusions (QDs) of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity.
We derive the fundamental generalized displacement solution using the Radon transform of stiffness based governing equations and present the direct formulation of the time-harmonic Boundary Element Method (BEM) for two-dimensional anisotropic and piezoelectric solids. The fundamental solution consists of the static singular and the dynamics regular parts; the former, evaluated analytically, is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle. The static BEM is a component of the time-harmonic BEM. We adopt the physical interpretation of Somigliana's identity to formulate the static BEM using the fundamental generalized point force and dislocation solutions obtained through the Stroh-Lekhnitskii (SL) formalism. We complete the time-harmonic BEM with the addition of the boundary integrals for the dynamic regular part which, from the original double integral representation, are reduced to simple line integrals along the unit circle.
We apply the BEM to the determination of the eigen frequencies of anisotropic and piezoelectric resonators. The eigenvalue problem deals with full non-symmetric complex-valued matrices whose components depend non-linearly on the frequency. We make a comparative study of non-linear eigenvalue solvers: QZ algorithm and the implicitly restarted Arnoldi method (IRAM). The FEM results whose accuracy is well established serve as the basis of the comparison. The IRAM is faster and has more control over the solution procedure than the QZ algorithm.
This is the first BEM for the eigenvalue analysis of the general anisotropic and piezoelectric solids in 2-D. The direct use of the fundamental solutions enables the accurate determination of eigenvalues.
The boundary element method (BEM) is developed for the plane problems of the isotropic elasticity based on a physical interpretation of Somigliana's identity. On the domain boundary two layers of different singularities -i.e., a layer of point forces and that of dislocation dipoles in an infinite domain- are placed to represent the effects of the boundary traction and the boundary displacements, respectively. A consistent use of the complex variables is made in the derivation of the Green's functions for the singularities and in closed form integration of the boundary integral terms after proper discretization is introduced. The method is especially useful in problems involving cracks, holes, inclusions, and so on which give rise to stress singularities or concentrations. For those problems the Green's functions are obtained analytically (based on the foundations on the BVP mentioned above) so that the desired boundary conditions can be automatically satisfied. As a result we do not have to try to simulate, for example, the stress singularity at the crack tip because, it is already built in the Green's functions.
In addition to the analytical Green's functions, numerical Green's functions and their hybrids are developed for the use as the fundamental solutions in the BEM formulation.
The whole crack singular element (WCSE) represents the analytical representation of the crack solution for straight center and edge cracks in the isotropic and anisotropic solids. The crack opening is represented by the continuous distribution of dislocation dipoles and the crack-tip singularity is embedded into the interpolation using orthogonal polynomials (i.e., Chebyshev and Jacobi) and their associated singular weight functions. The proposed analytical integration procedure of the Cauchy-type integrals defined over the crack eliminates the need for the quadrature formulae for numerical integration, streamlines, and enhances the accuracy of the traditional singular integral equation method for crack problems enabling the determination of the stress intensity factors in the main-processing without the need for the post-processing such as the use of J-integrals.
Simple analytical formulas for the displacement, traction, and the stress intensity factor for a 2-D crack tip singular element (CTSE) are developed using the continuous distribution of dislocation dipoles and Chebyshev polynomials. The known $\sqrt{r}$ crack tip opening displacement and the $1/ \sqrt{r}$ stress singularity at the crack tip is mathematically built in the formulas. In the boundary element implementation the crack tip singular element, which is a short whole crack singular element, is placed locally at each crack tip on top of the ordinary non-singular crack elements that cover the entire crack surface. Within the constraint of the rectilinear approximation, any curvilinear cracks can be modeled including center and edge cracks. Although the quarter point element is accurate and easy to use, it does not provide the analytical formula for the stress intensity factor which is the key feature of the proposed method.
We extend the singular crack tip element, developed for the 2D general anisotropic solids, to piezoelectric curvilinear multiple crack problems. The crack tip singularity is mathematically embedded in the BEM formulation to maintain the high numerical accuracy for the fast stress intensity factor calculation. Three crack surface electric boundary conditions ( impermeable, permeable, and semi-permeable) are tested for the multiple cracks configurations. The study extends the limitations of the existing work on the single straight crack to the general multiple curvilinear crack configurations and provides a reliable fracture analysis tool for real life calculation of the fracture of piezoelectric components widely used in industry.
[Sponsor: NSF INTERNATIONAL] The project will lead to the following developments: (1) A parallel computer program based on Fracture Mechanics and the Boundary Element Method for the numerical simulation of crack propagation in 2D solids. This program will be called: Fracture Processor. The Fracture Processor will be written in FORTRAN. (2) A parallel computer program based on Lumped Damage Mechanics, the Finite Element Method and the Boundary Element Method for the numerical simulation of damage and cracking in Reinforced Concrete structures subjected to dynamic loadings, including soil-structure interaction. This program will be called: Damage Processor. The Damage Processor will be written in FORTRAN. (3) WEB interfaces written in JAVA for data input for the fracture and damage processor: Pre-processor. (4) WEB interfaces written in JAVA for the visualization of the structural behavior, cracking propagation and damage evolution of 2D solids and framed buildings: Post-Processor. (5) Internet front ends for the programs specified in the objectives 1-4. (6) Hardware architectures of the lowest possible costs for the execution of many simulations.
[Sponsor: NSF INTERNATIONAL] We have developed a Java-based boundary element program front end, the Electronic Handbook of Fracture (e-Handbook), for fracture analysis of multiple curvilinear cracks in the two-dimensional general anisotropic solids. The e-Handbook provides a graphic user interface for the preparation of the input files, runs a Fortran program for the mixed-mode fracture analysis and displays the stress intensity factors at each crack tip. The e-Handbook is available for free download at http://rci.rutgers.edu/~denda/e-Handbook. The e-Handbook can be used as the electronic handbook of the stress intensity factors for custom two-dimensional crack configurations not available in the existing handbooks.
Boundary Element Analysis of Multiple Crack Growth in Elastic Solids
[With M.E. Marante, Lisandro Alvarado University, Department of Structural Engineering, Barquisimeto, Venezuela; University of the Andes, Department of Structural Engineering, Merida 5101, Venezuela ]
We have developed a boundary element code for the growth analysis of multiple cracks in the isotropic elastic solids. The crack opening displacement of each crack is represented by the distribution of dislocation dipoles along the crack line and the crack tip singular element (CTSE) is employed to embed the singular crack tip behavior mathematically. The dislocation dipole distribution is evaluated analytically to give a simple closed form formula for the CTSE. The CTSE provides an accurate yet fast algorithm for the crack growth analysis. In the presence of multiple cracks, the crack tip with the maximum stress intensity factor starts to grow once the critical stress intensity level is reached. Under the fixed applied load, we let this crack tip grow in the direction perpendicular to the maximum normal stress. During the incremental crack growth, we monitor the change in the stress intensity factors at all other crack tips including the current growing tip. For the current growing tip, the crack is allowed to grow as long as the stress intensity factor increases; if it decreases below the level of the critical stress intensity factor, then the crack growth is terminated. Other non-growing crack tips are allowed to grow if their stress intensity factors reach to the critical value. Similar crack growth analysis can be performed under other programmed loadings than the constant one. The proposed brittle crack growth analysis will provide the indispensable numerical tool to get insight into the effects of multiple crack interaction during their growth; such data have only been available experimentally so far.
A simplified two-dimensional elastic wing model is proposed to study the fluid-structure interaction effects of flexible wings in the flapping flight of insects. To incorporate the torsional deformation of the insect wings in the two-dimensional model, a narrow zone of reduced elastic stiffness is introduced in the model elastic wing to simulate the longitudinal flexion line. It is known that the longitudinal flexion line in the insect wing is the preferred axis of twist in the normal flight especially at the transition between down stroke and upstroke. In our two-dimensional model the narrow compliant zone is introduced at the leading edge of the wing, which also serves as the axis of twist as well as the steering axis of translational motion. The motion of this wing model in fluid is analyzed by our finite element method for the fluid-structure interaction which incorporates the interaction of the inertial force, fluid surface force, and the internal elastic force of the wing. The passive change of attack angle due to the interaction of these forces is predicted numerically. It is revealed that the wing keeps a high attack angle during translational portions of the stroke and rotates during stroke reversals. The fluid drag force and the elastic reaction given by the internal force take important role in keeping the high attack angle during translational portion of the stroke. Although the inertial force takes a dominant role in the rotation during the stroke reversal, the fluid drag force appears to assist the rotation. From the structural integrity point of view, the current wing model is inconsistent with the actual wing where the leading edge is highly stiffened. From the fluid-structure interaction point of view, our premise is that, despite this contradiction, the compliant zone model provides results remarkably in good agreement with those obtained by experiments by Dickinson et al. that incorporate the active change of attack angle. Results can be seen here.
References:In order to develop technologies to improve reliability of parts or tools from particle-reinforced metallic materials, the knowledge of the interrelations between microstructures of the materials and their fracture resistance is required. The purpose of the work is to develop an approach to optimize the distribution of hard inclusions in a matrix of the particle-reinforced materials in order to improve the fracture resistance of materials. Cast Al/Si alloys and tools steels are taken as test materials. Crack propagation in Al alloys and tool steels with different arrangements of hard inclusions (real and quasi-real idealized microstructures) is simulated using the boundary element method that employs the crack tip singular element developed by Denda. The force-displacement curves, fracture energy, fractal dimension and roughness of fracture surface are determined from the simulations. A parameter of the spatial inclusion distribution should be developed on the basis of the statistical analysis of the idealized inclusion arrangements. The parameter should be related to the numerically obtained fracture resistance of the ideal microstructures, and can serve as a basis for the improvement of real materials technologies. The concepts of the information theory are used to characterize and to determine optimal parameters of the inclusion arrangements. As a result, some recommendations for improvement of fracture resistance of the materials by varying their microstructures are developed.
Study of multiple-site ductile crack growth in frame structures under strong earthquake loading condition is carried out. The effects of plastic deformation, crack path meandering, and crack linkage are taken into account. The method of Continuous Distribution of Dislocations is used to represent curvilinear cracks and the Plastic Source Method is applied to describe the effects of plastic deformation. Reliable numerical algorithms for elastic-plastic and curvilinear multiple crack problems are being developed. Description of computer programs developed is given in Japanese.
[Sponsor: NSF IGERT] We represent the plastic deformation by the fundamental force dipole solution (plastic source method) and propose a technique to model the cyclic plastic deformation in the metallic glass. The plastic source method for the cyclic plastic deformation will be applied to model the fatigue crack growth. The crack tip stress field obtained will be compared with the experimental results obtained by the thermography technique developed by the University of Tennessee.