Dependent Boundary Value Research Articles

Multilevel continuation techniques for nonlinear boundary value problems with parameter dependence

A new technique is described to continue along solution branches of parameter- dependent nonlinear boundary value problems in order to obtain accurate multigrid solutions at specified points. In particular, the implementation for a general class of second order equations in divergence form in the program PLTMG is described and analyzed. This method has proven to be very robust and efficient, as is illustrated by several examples.

Overview no. 42 Texture development and strain hardening in rate dependent polycrystals

A new rate dependent constitutive model is developed for polycrystals subjected to arbitrarily large strains. The model is used to predict deformation textures and large-strain strain hardening behavior following various stress-strain histories for single phase f.c.c. aggregates that deform by crystallographic slip. Examples involving uniaxial and plane strain tension and compression are presented which illustrate how texture influences polycrystalline strain hardening, in particular these examples demonstrate both textural strengthening and softening effects. Input to the model includes the description of single crystal strain hardening and latent hardening along with strain rate sensitivity, all properties described on the individual slip system level. The constitutive formulation used for the individual grains is essentially that developed by Peirce et al. [6, Acta metall. 31, 1951 (1983)] to solve rate dependent boundary value problems for finitely deformed single crystals. Inclusion of rate dependence is shown to overcome the long standing problem of nonuniqueness in the choice of active slip systems which is inherent in the rate independent theory. Because the slipping rates on all slip systems within each grain are unique in the rate dependent theory, the lattice rotations and thus the textures that develop are unique. In addition, the model makes it possible to study how strain rate sensitivity on the slip system, and single grain, levels is manifested in polycrystalline strain rate sensitivity. The model is also used to predict “constant offset plastic strain yield surfaces” for materials that are nearly rate insensitive—these calculations describe the development of rounded “yield surface vertices” and the resulting softening of material stiffness to a change in loading path that vertices imply. For our rate dependent solid this reduction in stiffness occurs after small but finite loading increments. Finally the model is used to carry out an imperfection-based sheet necking analysis both for isotropic and strongly textured sheets. The results show that larger strain hardening rates, and strain rate sensitivity, on the slip system level both increase the failure strains, as expected, but also demonstrate a strong influence of texture on localized necking.

Some elementary solutions of the heat equation and related equations

An elementary connection is shown between certain time dependent boundary value problems for the heat equation and problems of steady state heat conduction (or diffusion) in a medium with a radially varying thermal conductivity. Solutions are given for growing ellipsoids, transient heating of wedges and transient bending of wedge shaped plates.

Solution of time dependent boundary value problems by the Boundary Operator Method

A new procedure, the Boundary Operator Method (BOM), for deriving normal mode response formulas in systems with time dependent boundary conditions is demonstrated. The method of analysis is applied to a finite, nonhomogeneous bar. This example illustrates the following advantages of the BOM: (1) The resulting formulas for the displacement and stress are valid for the entire history of the system and for Dirichlet, Neumann or Cauchy boundary conditions. (2) The uniqueness of the solution is unobscured since, unlike the Mindlin-Goodman method, the present solution technique does not require specific spatial functions which render the transformed boundary conditions homogeneous. (3) The evaluation of the solution is expedited by the absence of integrals containing auxiliary spatial functions. (4) The analysis employs standard mathematical techniques.