No abstract

Mixed weak formulations, with two or three main (tensor) variables, are stated and theoretically analyzed for general multi-dimensional dipolar Gradient Elasticity (biharmonic) boundary value problems. The general structure of constitutive equations is considered (with and without coupling terms). The mixed formulations are based on various generalizations of the so-called Ciarlet-Raviart technique. Hence, C 0 continuity conforming basis functions may be employed in the finite element approximations (or even, C À1 basis functions for the Cauchy stress variable). All the complicated boundary conditions, especially in the multi-dimensional scenario, are naturally considered. The main variables are the displacement vector, the double stress tensor and the Cauchy stress tensor. The latter variable may be eliminated in some of the formulations, depending on the structure of the constitutive equations. The standard continuous and discrete Babuška-Brezzi inf-sup conditions for the constraint equation, as well as, solution uniqueness for both the continuous statements and discrete approximations, are established in all cases. For the purpose of completeness, two one-dimensional mixed formulations are also analyzed. The respective constitutive equations possess general structure (with coupling terms). For the 1-D formulations, all the inf-sup conditions are satisfied, for both the continuous and discrete statements (assuming proper selection of the polynomial spaces for the main variables). Hence, the general Babuška-Brezzi theory results in quasi-optimality and stability. For multi-dimensional problems, the difficulty of deducing the inf-sup condition on the kernel is examined. Certain aspects of methodologies employed to theoretically by-pass this problem, are also discussed.

SUMMARYAn adaptive global-local refinement strategy based on the interior error estimates of the h-method is proposed. Adaptive global-local refinement strategy is aimed at constructing nearly optimal finite element meshes, where the force transfer to the local region of interest is sufficiently accurate so that the local phenomena of interest is resolved with a user-specified accuracy. Numerical examples for linear elasticity problems in two dimensions together with a comparison to the classical adaptive h-refinement strategy based on the equidistribution of errors are presented to validate the present formulation.

SUMMARYA methodology has been developed to accurately resolve the stress field in the vicinity of free edges as well as the overall response of laminated plates without significantly affecting the computational cost. This is accomplished by enriching a set of classical smooth interpolants throughout the thickness direction with Co continuous displacement interpolants (piecewise continuous strain field) in the regions where the most critical behaviour is anticipated. C o continuity of the displacement field is maintained by imposing homogeneous boundary conditions on the superimposed field in the portion of the boundary which is not contained within the boundary of the problem. Numerical experiments for both cylindrical bending and uniform extension of cross-ply laminates are presented to validate the present formulation.

Mixed formulations with C 0 -continuity basis functions are employed for the solution of some types of one-dimensional fourth-and sixth-order equations, resulting from axial tension and buckling of gradient elastic beams, respectively. A basic characteristic of gradient elasticity type equations is the appearance of boundary layers in the higher-order derivatives of the displacements (e.g., in the stress fields). This is due to the small parameters (related to the size of the microstructure) entering the governing equations. The proposed mixed formulations are based on generalizations of the well-known Ciarlet-Raviart mixed method, where the new main variables are related to second-order (or fourth order, for the buckling problem) derivatives of the displacement field. The continuous and discrete Babuška-Brezzi inf-sup conditions are established. The mixed formulations are numerically tested for both the uniform h-and p-extensions. With regard to the axial tension problem, the standard quasi-optimal rates of convergence are numerically verified in all cases (i.e., algebraic rate of convergence for the h-extension and exponential rate for the p-extension). On the other hand, the h-extension observed convergence rates of the critical (buckling) load for the second model problem are slightly higher than the theoretical ones found in the literature (especially for polynomial order p = 1). The respective observed rates of convergence of the buckling load for the p-extension are still exponential.