Displacement from boundary point forces

As a result of applying a point force on the boundary, displacements in the semiinfinite solid are produced. These displacements can be calculated from Hooke s law and the displacement-strain relationships. The displacement in the r-direction, /r, is given by Eq. (2.20) and Eq. (2.23) as... 

The influence of this pressure distribution on the displacements in a neighboring area on the same boundary surface can be analyzed as shown in Fig. 2.8. Let M be an arbitrary point outside the circular area (Fig. 2.8(a)) or inside the circular area (Fig. 2.8(b)) with a distance r from the center of the circular area. A small element within the loaded zone is chosen as s d/5 ds, where s is the distance between this element and M. From the solution of lz for a point force on a semiinfinite solid, the increment of the vertical displacement at location M under the influence of this element force would be... 

The components of represent stochastic displacements and are obtained using the multivariate Gaussian random number generator GGNSM from the IMSL subroutine library (30). p ° is the initial hydrodynamic interaction tensor between subunits iJand j. Although the exact form of D. is generally unknown, it is approximated here using the Oseen tensor with slip boundary conditions. This representation has been shown to provide a reasonable and simple point force description of the relative diffusion of finite spheres at small separations (31). In this case, one has... 

The sum over all the elements will yield all the components of the nodal forces at every nodal point, including the externally applied nodal forces and the reactive forces at the boundary due to displacement-specified boundary conditions. The calculation of nodal forces using eqn.(24) is an exact treatment in the sense that it has nothing to do with the approximations whatsoever involved in obtaining the incremental displacements, strains, and stresses. Therefore, the numerical results show that the total forces and the total moments due to the calculated nodal forces are all vanishing the computer software passes another test. However, the calculated nodal forces may turn out to be different from those expected and, when that happens, the differences will serve as the forcing terms in the next iteration until the differences are within the specified error tolerance. 

Numerical solutions are also possible and in view of the complexity of the analytical solutions often desirable. One method is to replace the Lame equations (16) by a set of difference equations for points on an array over an r-z section of the cylinder. The boundary conditions are then used to remove undefined points at the boundaries. The solution is obtained by iteration (23) or by solving the equations directly by a matrix technique (24). The other common method is the use of finite elements, which have been applied widely to axisymmetric elastic and thermoelastic problems (23). This technique breaks the r-z section of the cylinder into regions or elements where the properties and conditions can be assumed to be approximately uniform. At the junctions of the elements, the nodes, displacements, and forces are defined. These displacements and forces are connected, using the elastic and thermal properties of the material, by minimizing the energy of the system. A set of linear equations in terms of the displacements is then obtained by matching the nodal forces and displacements from element to element, together with the boundary conditions. The set of linear equations is then solved in the same way as for the finite difference approximation. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

---