E. Schnack

Hybrid coupled finite–boundary element methods for elliptic systems of second order

The work of G.C. Hsiao was carried out during several visits at the Universitat at Stuttgart supported by the DFG priority research programme “Boundary Element Methods” within the guest–programme We–659/19–2,3; the work of E. Schnack was supported by the DFG projects Schn–245/3–1,2,3 and Schn–245/4–1,2, by the DFG priority research programme “Boundary Element Methods” within the the projects Schn– 245/10–1,2,3, Schn–245/17–1 and the MIMD–DD–Project Schn–245/20–1; the work of W.L. Wendland was supported during several visits at the University of Delaware by the Fulbright Commission, the University of Delaware and the Delaware MURI–grant. The final version of the work was completed during G.C. Hsiao’s visit at University of Stuttgart in the summer 1997 supported by the Alexander von Humboldt–Stiftung.

A hybrid coupled finite-boundary element method in elasticity

Abstract We present a hybrid coupled finite-boundary element method for a mixed boundary-value problem in linear elasticity. In this hybrid method, we consider, in addition to traditional finite elements, the Trefftz elements for which the governing equations of equilibrium are required to be satisfied. The Trefftz elements are then modelled with boundary potentials supported by the individual element boundaries, the so-called macro-elements. The coupling between the finite and macro-elements is accomplished by using a generalized compatibility condition in weak sense with mortar elements on the skeleton. The latter allows us to relax the continuity requirements for the global displacement field. In particular, the mesh points of the macro-elements can be chosen independently of the nodes of the FEM structure. This approach permits the combination of independent meshes and also the exploitation of modern parallel computing facilities. By following Hsiao et al. [G.C. Hsiao, E. Schnack and W.L. Wendland, Hybrid coupled finite-boundary element methods for elliptic systems of second order, in preparation], we give the precise formulation of the method, its functional analytic setting as well as corresponding discretizations and asymptotic error estimates. For illustrations, some computational results are also included.

Singularities near three-dimensional corners in composite laminates

The high interlaminar stresses, which appear in laminated composites due to the boundary layer effect near the free edge, play an important role in the analysis and design of advanced structures. Moreover, they are also the dominant effect causing delamination. Even if the singular behavior of such structures is investigated in many works, most of them deal either with 2D, or with pseudo-3D problems, i.e. problems of two variables in a three-dimensional space. However, some numerical and experimental findings indicate that laminated plates exhibit a tendency to delaminate at corners, an effect impossible to be determined by a two-dimensional analysis. The aim of the present paper is to investigate stress singularities in a laminated composite wedge under consideration of real three-dimensional corner effects. A weak formulation, as well as a finite element approximation technique introduced in the past for isotropic problems is extended here to cover anisotropic material properties. This formulation leads to a quadratic eigenvalue problem, which is solved iteratively using the Arnoldi method. The first singular terms in the asymptotical expansion of the linear-elastic solution near the vertex of the wedge are obtained as eigenpairs of this eigenvalue problem. The order and mode of singularity are reported for all wedge angles and different fiber orientations for angle and cross-ply laminates. All calculations are based on a typical for some high modulus graphite-epoxy systems orthotropic material model.

Micromechanical Modeling of Porous Carbon/Carbon Composites

A procedure to model fiber-reinforced composites containing pores of irregular shapes is presented. Closed-form expressions for contributions of fibers and pores into effective elastic moduli are provided. The procedure is applied to predict the transverse elastic properties of unidirectional carbon/carbon composites (carbon fibers in pyrolytic carbon matrix) densified by chemical vapor infiltration. Infiltration treatment results in the formation of irregularly shaped pores randomly oriented in the plane perpendicular to the direction of fiber (transverse plane). These pores are analyzed using a numerical conformal mapping technique, and their contribution to the effective elastic properties is expressed in terms of the cavity compliance contribution tensor. Components of this tensor are found for a variety of typical pore shapes.

An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body

This paper deals with the inverse problem of determining cracks lying on an a priori known internal surface inside a three-dimensional elastic body from overdetermined elastostatic boundary data on the outer surface. A new alternating iterative algorithm for the reconstruction of such internal cracks is proposed with two variants: for the first variant the complete Cauchy data of one measurement have to be given on the whole outer boundary, whereas the other variant only needs overdetermined boundary data on some part of the outer boundary. With this approach, the case of multiple and generally shaped cracks lying on a curved internal surface inside an anisotropic material can also easily be treated. Numerical examples are given for the case of a planar elliptic crack in an isotropic elastic body, demonstrating the fast convergence and good regularizing properties of the proposed algorithm. Due to the general character of the algorithm, it can also easily be extended to other elliptic operators.

Some fundamental solutions for the Kirchhoff, Reissner and Mindlin plates and a unified BEM formulation

Abstract We derive analytical solutions for the deflection of thin circular plates, which are loaded by centrally located concentrated bending moments and transverse forces. Greens functions for clamped and simply supported plates are presented. Reduction of these Greens functions leads to the corresponding fundamental solution for the Kirchhoff plate bending model (K problem). This fundamental solution reduces to those obtained through the direct simplification of the fundamental solution for the sixth-order Reissner and Mindlin plate bending models (RM problem). This allows to decompose each fundamental tensor of the problem RM into the sum of the fundamental tensor of the problem K and a correction tensor (Sh problem), which contains the contribution of the shear strains, e.g. UijRM(r)=UijK(r)+UijSh(r). Within the boundary element analysis this enables the investigation of the contributions of the shear strains to the solutions of Reissner and Mindlin plate bending models, as corrections of the Kirchhoff values, all determined from the same BEM code. This opens up new possibilities for the analysis of plates by the BEM.

The Vibration Analysis of FGM Truncated Conical Shells Resting on Two-Parameter Elastic Foundations

An analytical formulation is presented for the vibration analysis of truncated conical shells made of functionally graded material and resting on the Winkler-Pasternak foundations. It is assumed that the truncated conical shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The governing equations according to the Donnells theory are solved by Galerkins method and the fundamental frequencies with or without two-parameter elastic foundation have been found. The effects of the elastic foundations, changing large radius-to-small radius ratio, lengths-to-radius ratio, material composition and volume fraction of constituent materials on the fundamental frequencies of the truncated conical shell are investigated.

Hierarchical material modeling of carbon/carbon composites

Unidirectional, long fiber carbon/carbon composites fabricated by chemical vapor infiltration (CVI) consisting of carbon fibers in a pyrolytic carbon matrix are anisotropic materials. It is practically impossible to identify experimentally the elastic properties (modules) of this anisotropic material. The aim of this investigation is to predict the elastic properties of this composite theoretically. The study of this material with the help of microscopy gives information about the very complicated anisotropic structure of this composite at each length scale. That is the reason that a hierarchical model for this material is developed, which consists of four length levels. A methodology for identification of the elastic properties for such composites is proposed. The problem is solved with the help of a homogenization procedure for each level.

Shape optimization under fatigue using continuum damage mechanics

An introduction and overview of the developments in shape optimization for fatigue problems is given, beginning with the main ideas in the field of static shape optimization. The use of numerical tools together with the incorporation of non-linear material properties in the continuum mechanics design has led to new possibilities for the analysis of fatigue behaviour in mechanical engineering structures. Based on publications by Lemaitre and Chaboche, the material behaviour is described by defining different partial differential equation systems on micro-scale and meso-scale. From this an optimal algorithm results for the shape optimization of mechanical engineering structures using continuum damage mechanics. Our numerical and experimental tests show a significant increase of lifetime in comparison to classical shapes.

Control of the Von Mises Stress with Dynamic Programming

The problem treated is the minimization of the von Mises equivalent stress in elastostatic structures subjected to manufacturing and behavioural constraints by variation of partial boundaries. This constrained non-linear problem is solved using a discrete non-gradient approach which allows a high number of design variables and works without any restrictions imposed to the geometrical shape of the design boundary. The existence of the solution for this class of problems follows from a dynamic programming formulation. Axisymmetric and three-dimensional problems are solved using this method.