Elastostatics

In this chapter, we are going to describe the main features as well as the basic steps of the Boundary Element Method (BEM) as applied to elastostatic problems and to compare them with other numerical procedures. As we shall show, it is easy to appreciate the adventages of the BEM, but it is also advisable to refrain from a possible unrestrained enthusiasm, as there are also limitations to its usefulness in certain types of problems. The number of these problems, nevertheless, is sufficient to justify the interest and activity that the new procedure has aroused among researchers all over the world. Briefly speaking, the most frequently used version of the BEM as applied to elastostatics works with the fundamental solution, i.e. the singular solution of the governing equations, as an influence function and tries to satisfy the boundary conditions of the problem with the aid of a discretization scheme which consists exclusively of boundary elements. As in other numerical methods, the BEM was developed thanks to the computational possibilities offered by modern computers on totally "classical" basis. That is, the theoretical grounds are based on linear elasticity theory, incorporated long ago into the curricula of most engineering schools. Its delay in gaining popularity is probably due to the enormous momentum with which Finite Element Method (FEM) penetrated the professional and academic media. Nevertheless, the fact that these methods were developed before the BEM has been beneficial because de BEM successfully uses those results and techniques studied in past decades. Some authors even consider the BEM as a particular case of the FEM while others view both methods as special cases of the general weighted residual technique. The first paper usually cited in connection with the BEM as applied to elastostatics is that of Rizzo, even though the works of Jaswon et al., Massonet and Oliveira were published at about the same time, the reason probably being the attractiveness of the "direct" approach over the "indirect" one. The work of Tizzo and the subssequent work of Cruse initiated a fruitful period with applicatons of the direct BEM to problems of elastostacs, elastodynamics, fracture, etc. The next key contribution was that of Lachat and Watson incorporating all the FEM discretization philosophy in what is sometimes called the "second BEM generation". This has no doubt, led directly to the current developments. Among the various researchers who worked on elastostatics by employing the direct BEM, one can additionallly mention Rizzo and Shippy, Cruse et al., Lachat and Watson, Alarcón et al., Brebbia el al, Howell and Doyle, Kuhn and Möhrmann and Patterson and Sheikh, and among those who used the indirect BEM, one can additionally mention Benjumea and Sikarskie, Butterfield, Banerjee et al., Niwa et al., and Altiero and Gavazza. An interesting version of the indirct method, called the Displacement Discontinuity Method (DDM) has been developed by Crounh. A comprehensive study on various special aspects of the elastostatic BEM has been done by Heisse, while review-type articles on the subject have been reported by Watson and Hartmann. At the present time, the method is well established and is being used for the solution of variety of problems in engineering mechanics. Numerous introductory and advanced books have been published as well as research-orientated ones. In this sense, it is worth noting the series of conferences promoted by Brebbia since 1978, wich have provoked a continuous research effort all over the world in relation to the BEM. In the following sections, we shall concentrate on developing the direct BEM as applied to elastostatics.

Discontinuous solutions and boundary value problems for non-linearly elastic fibre-reinforced materials

En este trabajo se han analizado varios problemas en el contexto de la elasticidad no lineal basándose en modelos constitutivos representativos. En particular, se han analizado problemas relacionados con el fenómeno de perdida de estabilidad asociada con condiciones de contorno en el caso de material reforzados con fibras. Cada problema se ha formulado y se ha analizado por separado en diferentes capítulos. En primer lugar se ha mostrado el análisis del gradiente de deformación discontinuo para un material transversalmente isótropo, en particular, el modelo del material considerado consiste de una base neo-Hookeana isótropa incrustada con fibras de refuerzo direccional caracterizadas con un solo parámetro. La solución de este problema se vincula con instabilidades que dan lugar al mecanismo de fallo conocido como banda de cortante. La perdida de elipticidad de las ecuaciones diferenciales de equilibrio es una condición necesaria para que aparezca este tipo de soluciones y por tanto las inestabilidades asociadas. En segundo lugar se ha analizado una deformación combinada de extensión, inación y torsión de un tubo cilíndrico grueso donde se ha encontrado que la deformación citada anteriormente puede ser controlada solo para determinadas direcciones de las fibras refuerzo. Para entender el comportamiento elástico del tubo considerado se ha ilustrado numéricamente los resultados obtenidos para las direcciones admisibles de las fibras de refuerzo bajo la deformación considerada. En tercer lugar se ha estudiado el caso de un tubo cilíndrico grueso reforzado con dos familias de fibras sometido a cortante en la dirección azimutal para un modelo de refuerzo especial. En este problema se ha encontrado que las inestabilidades que aparecen en el material considerado están asociadas con lo que se llama soluciones múltiples de la ecuación diferencial de equilibrio. Se ha encontrado que el fenómeno de instabilidad ocurre en un estado de deformación previo al estado de deformación donde se pierde la elipticidad de la ecuación diferencial de equilibrio. También se ha demostrado que la condición de perdida de elipticidad y ^W=2 = 0 (la segunda derivada de la función de energía con respecto a la deformación) son dos condiciones necesarias para la existencia de soluciones múltiples. Finalmente, se ha analizado detalladamente en el contexto de elipticidad un problema de un tubo cilíndrico grueso sometido a una deformación combinada en las direcciones helicoidal, axial y radial para distintas geotermias de las fibras de refuerzo . In the present work four main problems have been addressed within the framework of non-linear elasticity based on representative constitutive models. Namely, problems related to the loss of stability phenomena associated with boundary value problems for fibre-reinforced materials. Each of the considered problems is formulated and analysed separately in different chapters. We first start with the analysis of discontinuous deformation gradients for a transversely isotropic material under plane deformation. In particular, the material model is an augmented neo-Hookean base with a simple unidirectional reinforcement characterised by a single parameter. The solution of this problem is related to material instabilities and it is associated with a shear band-type failure mode. The loss of ellipticity of the governing differential equations is a necessary condition for the existence of these material instabilities. The second problem involves a detailed analysis of the combined non-linear extension, inflation and torsion of a thick-walled circular cylindrical tube where it has been found that the aforementioned deformation is controllable only for certain preferred directions of transverse isotropy. Numerical results have been illustrated to understand the elastic behaviour of the tube for the admissible preferred directions under the considered deformation. The third problem deals with the analysis of a doubly fibre-reinforced thickwalled circular cylindrical tube undergoing pure azimuthal shear for a special class of the reinforcing model where multiple non-smooth solutions emerge. The associated instability phenomena are found to occur prior to the point where the nominal stress tensor changes monotonicity in a particular direction. It has been also shown that the loss of ellipticity condition that arises from the equilibrium equation and ^W=2 = 0 (the second derivative of the strain-energy function with respect to the deformation) are equivalent necessary conditions for the emergence of multiple solutions for the considered material. Finally, a detailed analysis in the basis of the loss of ellipticity of the governing differential equations for a combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube is carried out.

Detección de grietas en placas con técnicas multifrecuencia

Safety is one of the most important feature in the aviation industry, and this involves too many factors. One of these is the aircraft maintenance. Over time, the procedures have been changing, and improving themselves. Non Destructive Testing (NDT) appeared in the late 19th century as a great option, because it enabled to inspect any structure without damaging it. Nowadays, there are several kinds of NDT, but ultrasound is one of the most widely used. This Master Thesis is devoted to an innovative ultrasound technique for crack detection. A technique, whose main aim lies in getting a good location of defects from a few measures, breaking with the currently widespread methods, as phased array. It is not necessary to use trains of waves, only discrete excitations, which means a great saving of time and energy. This work is divided into two steps: the first is to develop a multiphysics simulator, which is able to solve linear elasticity 3D problems (via Finite Element Method, FEM). This simulator allows to obtain in a computationally efficient way the displacement field for different frequencies and excitations. The solution of this elastic problem is needed to be used in the second step, which consists of generating a code that implements a mathematical tool named topological derivative, allowing to locate defects in the studied domain. In this work, the domain is a plate, and the defect is a hidden spherical void. The simulator has been developed using open source software (Elmer, Gmsh, ...), achieving a highly versatile simulator, which allows to change the configuration easily: domain size and shape, number and position of transducers, etc. Just one comercial software is used, Matlab. It is used to implement the topological derivative. In this work, the performance of the method is tested in several examples comparing the results when one or more frequencies are considered for different configurations of emisors/receptors.

Application of the Boundary Method to the determination of the properties of the beam cross-sections

Using the 3-D equations of linear elasticity and the asylllptotic expansion methods in terms of powers of the beam cross-section area as small parameter different beam theories can be obtained, according to the last term kept in the expansion. If it is used only the first two terms of the asymptotic expansion the classical beam theories can be recovered without resort to any "a priori" additional hypotheses. Moreover, some small corrections and extensions of the classical beam theories can be found and also there exists the possibility to use the asymptotic general beam theory as a basis procedure for a straightforward derivation of the stiffness matrix and the equivalent nodal forces of the beam. In order to obtain the above results a set of functions and constants only dependent on the cross-section of the beam it has to be computed them as solutions of different 2-D laplacian boundary value problems over the beam cross section domain. In this paper two main numerical procedures to solve these boundary value pf'oblems have been discussed, namely the Boundary Element Method (BEM) and the Finite Element Method (FEM). Results for some regular and geometrically simple cross-sections are presented and compared with ones computed analytically. Extensions to other arbitrary cross-sections are illustrated.

Optimal design of a beam subjected to compression forces in the framework of different structural models

The optimal design of a vertical cantilever beam is presented in this paper. The beam is assumed immersed in an elastic Winkler soil and subjected to several loads: a point force at the tip section, its self weight and a uniform distributed load along its length. lbe optimal design problem is to find the beam of a given length and minimum volume, such that the resultant compressive stresses are admisible. This prohlem is analyzed according to linear elasticity theory and within different alternative structural models: column, Navier-Bernoulli beam-column, Timoshenko beamcolumn (i.e. with shear strain) under conservative loads, typically, constant direction loads. Results obtained in each case are compared, in order to evaluate the sensitivity of model on the numerical results. The beam optimal design is described by the section distribution layout (area, second moment, shear area etc.) along the beam span and the corresponding beam total volume. Other situations, some of them very interesting from a theoretical point of view, with follower loads (Beck and Leipholz problems) are also discussed, leaving for future work numerical details and results.

Boundary elements in potential and elasticity theory

A general theory that describes the B.I.E. linear approximation in potential and elasticity problems, is developed. A method to tread the Dirichlet condition in sharp vertex is presented. Though the study is developed for linear elements, its extension to higher order interpolation is straightforward. A new direct assembling procedure of the global of equations to be solved, is finally showed.