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.
The paper examines the nonlinear elastic-plastic response of internally pressurized 90 deg pipe elbows under in-plane and out-of-plane bending. Nonlinear shell elements from a general-purpose finite element program are employed to model the inelastic response of steel elbows and the adjacent straight parts. The numerical results are successfully compared with real-scale experimental measurements. The paper also presents a parametric study, aimed at investigating the effects of diameter-to-thickness ratio and moderate pressure levels on the ultimate bending capacity of 90 deg elbows, focusing on the failure mode (local buckling or cross-sectional flattening) and the maximum bending moment. Special attention is given to the response of 90 deg elbows under out-of-plane bending moments.
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.
The goal of this work is a systematic presentation of some classes of mixed weak formulations, for general multi-dimensional dipolar gradient elasticity (fourth order) boundary value problems. The displacement field main variable is accompanied by the double stress tensor and the Cauchy stress tensor (case 1 or l À s À u formulation), the double stress tensor alone (case 2 or l À u formulation), the double stress, the Cauchy stress, the displacement second gradient and the standard strain field (case 3 or l À s À j À e À u formulation) and the displacement first gradient, along with the equilibrium stress (case 4 or u À h À c formulation). In all formulations, the respective essential conditions are built in the structure of the solution spaces. For cases 1, 2 and 4, one-dimensional analogues are presented for the purpose of numerical comparison. Moreover, the standard Galerkin formulation is depicted. It is noted that the standard Galerkin weak form demands C 1 -continuous conforming basis functions. On the other hand, up to first order derivatives appear in the bilinear forms of the current mixed formulations. Hence, standard C 0 -continuous conforming basis functions may be employed in the finite element approximations. The main purpose of this work is to provide a reference base for future numerical applications of this type of mixed methods. In all cases, the associated quadratic energy functionals are formed for the purpose of completeness.
A mixed formulation is developed and numerically validated for the general 2D anti-plane shear problem in micro-structured solids governed by dipolar strain gradient elasticity. The current mixed formulation employs the form II statement of the gradient elasticity theory and uses the double stress components and the displacement field as main variables. High order, C 0 -continuous, conforming basis functions are employed in the finite element approximations ( p-version). The results for the mode III crack problem reveal that, with proper mesh refinement at the areas of high solution gradients, the current approximation method captures the exact solution behaviour at different length scales, which depend on the size of material micro-structure. The latter is of vital importance because, near the crack tip, the nature of the exact solution, changes radically as we proceed from the macro-to micro-scale.
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