Theoretical and computational concepts in engineering mechanics

Traditional finite element methods and boundary element methods cannot accurately analyze non-smooth problems in engineering continuum mechanics. In non-smooth problems, either the domains have sharp boundaries including cracks and comers or mixed boundary conditions are specified. The common feature of these problems is singularity. In engineering applications, non-smooth problems include those associated with fracture mechanics, and also those involving composite materials in which interface conditions may include singularities. Solution of these problems are crucial to the design of safe, efficient components.;In this dissertation, a new theory for boundary value problems has been developed. This theory which is called the theory of fundamental eigen-expansion , introduces fundamental deformations as a basis for general solution. This theory not only explains direct integral equation formulations and BEM, but also gives a new formulation for non-smooth problems.;The theory of fundamental eigen-expansion uses the concept of orthogonal functions in boundary value problems. We prove that the fundamental eigenmodes exist and are orthogonal with respect to an arbitrary weight function. Using the appropriate weight function is the key to solving non-smooth problems. These eigenmodes are the spectrum of the direct integral equation.;The new theory gives two convergence criteria. One is global convergence and the other is local. Global convergence brings the concept of "convergence in mean" which is a popular property in the theory of orthogonal functions. Local convergence relates to the idea of "uniform convergence" and explains the behavior of solutions for discontinuity on the boundary, which is a common feature of non-smooth problems.;In this dissertation, a traction oriented FEM formulation is also developed which accounts for singularities. This formulation follows the theory of fundamental eigen-expansion. By condensation for internal nodes we can have an exact correspondence to the BEM equations. The FEM equation deals with symmetric matrices which shows that the spectrum is real. The theory of fundamental eigen-expansion shows the relation of BEM and the new FEM. We can say that the new FEM is an approximated BEM where volume integral is transformed on the boundary numerically by condensation.;Beyond its usefulness in solving non-smooth problems, the theory of fundamental eigen-expansion gives a better understanding of boundary value problems which directly affects computational mechanics methods such as BEM and FEM.