A triangular hybrid equilibrium plate element of general degree

Hybrid stress‐based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

In this paper, we extend to Reissner‐Mindlin plate bending problems a technique, originally proposed in the context of two‐dimensional and three‐dimensional continua, for recovering fully equilibrated stresses from the solution of a compatible finite element model. The technique involves partition of unity functions and the analyses of overlapping star patches modelled with hybrid equilibrium plate elements. The patches are subjected to balanced systems of loads composed of partitioned and fictitious loads, where the latter are derived from the stresses of the compatible solution. The special case of assumed linear displacement fields of both deflection and rotation for the compatible model is included. This case requires additional fields of stress resultants to correct possible rotational imbalances of star patches, and these are derived elementwise. Other cases of nonconforming elements are briefly considered. Numerical examples are presented to illustrate the effectiveness of these techniques in terms of the deviation of the recovery, which compares the complementary strain energy of a recovered solution with that obtained by a global equilibrated analysis based on the same stress approximations.     

The influence of dietary fatty acid composition on N-ethyl-N-nitrosourea-induced mammary tumour incidence in the rat and on the composition of inositol- and ethanolamine-phospholipids of normal and tumour mammary tissue

This study has investigated the influence of dietary fatty acid composition on mammary tumour incidence in N-ethyl-N-nitrosourea (ENU)-treated rats and has compared the susceptibility to dietary fatty acid modification of the membrane phospholipids phosphatidylinositol (PI) and phosphatidylethanolamine (PE) from normal and tumour tissue of rat mammary gland. The incidence of mammary tumours was significantly lower in fish oil--(29%), compared with olive oil--(75%; P < 0.04) but not maize oil--(63%; P < 0.1) fed animals. No differences in PI fatty acid composition were found in normal or tumour tissue between rats fed on maize oil, olive oil or fish oil in diets from weaning. When normal and tumour tissue PI fatty acids were compared, significantly higher amounts of stearic acid (18:0) were found in tumour than normal tissue in rats given olive oil (P < 0.05). A similar trend was found in animals fed on maize oil, although differences between normal and tumour tissue did not reach a level of statistical significance (P < 0.1). In mammary PE, maize oil-fed control animals had significantly higher levels of linoleic acid (18:2n-6) than either olive oil- or fish oil-fed animals (P < 0.05, both cases) and levels of arachidonic acid were also higher in maize oil- compared with fish oil-fed animals (P < 0.05). In tumour-bearing animals no differences in PE fatty acid composition were found between the three dietary groups. When normal and tumour tissue PE fatty acids were compared, significantly lower amounts of linoleic acid (18:2n-6; P < 0.01) and significantly greater amounts of arachidonic acid (20:4n-6; P < 0.05) were found in tumour than normal tissue of rats fed on maize oil. The present study shows that the fatty acid composition of PI from both normal and tumour tissue of the mammary gland is resistant to dietary fatty acid modification. The PE fraction is more susceptible to dietary modification and in this fraction there is evidence of increased conversion of linoleic acid to arachidonic acid in tumour compared with normal tissue. Lower tumour incidence rates in rats given fish oils may in part be due to alteration in prostanoid metabolism secondary to displacement of arachidonic acid by eicosapentaenoic acid, but PE rather than PI would appear to be the most likely locus for diet-induced alteration in prostanoid synthesis in this tissue. Effects of dietary fatty acids other than on the balance of n-6 and n-3 fatty acids, and on prostanoid metabolism, should also be considered. The significance of increased stearic acid content of PI in tumours of olive oil-fed animals and the possible influence of dietary fatty acids on the capacity for stearic acid accumulation requires further study.     

Recovering local equilibrium from three-dimensional compatible finite element solutions

We study the application to compressible and incompressible three-dimensional elasticity problems of the technique that we proposed in 2009 for the recovery of equilibrated stresses from compatible finite element solutions. The case of finite elements with linear displacement approximations, for which the partitioned systems of loads are not initially balanced in terms of rotational equilibrium, addressed in 2012, is also considered. The article includes a presentation of the implementation details, a study of the convergence in terms of the energy of the error, and a characterization of the solutions obtained.     

A general degree hybrid equilibrium finite element for Kirchhoff plates

We present a family of hybrid equilibrium finite elements for the quasistatic linear elastic analysis of plates governed by Kirchhoff theory. The moments are approximated by self‐balanced polynomial fields of general degree, and in order to impose strong codiffusivity, the normal boundary rotations are approximated with complete polynomials of the same degree, whereas the transverse deflections use polynomials one degree lower. Furthermore, it is also necessary to include an independent approximation of the vertex translations. We show that the triangular form of this element is stable, that is, free from spurious kinematic modes, and the formulation that we present allows these elements to be used as a standard displacement element. Examples of computed values and convergence of the solutions are presented, which demonstrate the performance of these elements. Copyright © 2013 John Wiley & Sons, Ltd.