Theoretical analysis of a class of mixed, C0 continuity formulations for general dipolar Gradient Elasticity boundary value problems

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.

A mixed finite volume formulation for the solution of gradient elasticity problems

“…In the following lines, the equations of the general multidimensional dipolar gradient elasticity formulation in Forms I and II are presented (see also [15]):…”

“…Special focus is also given in the numerical solution of strain gradient equations via finite and boundary element methods. Tsamasphyros [21,22], Markolefas [15], Amanatidou [1], Borst [3,4] are some of the many researchers working towards this direction.…”

Mixed finite element formulation for the general anti-plane shear problem, including mode III crack computations, in the framework of dipolar linear gradient elasticity

“…Assuming isotropic constitutive relations (hence, there are no coupling terms between standard and double stresses [13,14]), the following stress-strain relations may be stated for the non-vanishing stress components in a given Cartesian coordinate system x, y, z, Standard (monopolar) Cauchy stresses:…”

“…Therefore, for the purpose of generality, we currently consider them as independent variables. Using (2.2), (2.3) and the equilibrium equations of the dipolar gradient elasticity theory [4,13,15], the following fourth order governing differential equation is derived, see [7] for details,…”

“…The main variables are the double stress tensor and the displacement field. The current formulation falls in the general category of mixed methods developed in [13], see also [3,12,18]. The solution function spaces are appropriately defined and the final weak forms have the standard (symmetric) mixed structure.…”