A control volume-based finite element method for plane micropolar elasticity

This paper describes the development of a numerical procedure for predicting deformations and stresses in a loaded two-dimensional membrane exhibiting micropolar or Cosserat constitutive behaviour. The procedure employs a conventional finite element (FE) mesh together with a dual mesh of interconnected control volumes, each of which must satisfy equilibrium. A series of patch tests covering a variety of simple strain states are used to validate the procedure that is then employed to predict the stress concentration in a membrane containing a small hole. The predictions provided by the procedure are compared with those given previously by FEs

A higher order control volume based finite element method to prodict the deformation of heterogeneous materials

Materials with obvious internal structure can exhibit behaviour, under loading, that cannot be described by classical elasticity. It is therefore important to develop computational tools incorporating appropriate constitutive theories that can capture their unconventional behaviour. One such theory is micropolar elasticity. This paper presents a linear strain control volume finite element formulation incorporating micropolar elasticity. Verification results from a micropolar element patch test as well as convergence results for a stress concentration problem are included. The element will be shown to pass the patch test and also exhibit accuracy that is at least equivalent to its finite element counterpart

The influence of void size on the micropolar constitutive properties of model heterogeneous materials

In this paper the mechanical behaviour of model heterogeneous materials consisting of regular periodic arrays of circular voids within a polymeric matrix is investigated. Circular ring samples of the materials were fabricated by machining the voids into commercially available polymer sheet. Ring samples of differing sizes but similar geometries were loaded using mechanical testing equipment. Sample stiffness was found to depend on sample size with stiffness increasing as size reduced. The periodic nature of the void arrays also facilitated detailed finite element analysis of each sample. The results obtained by analysis substantiate the observed dependence of stiffness on size. Classical elasticity theory does not acknowledge this size effect but more generalized elasticity theories do predict it. Micropolar elasticity theory has therefore been used to interpret the sample stiffness data and identify constitutive properties. Modulus values for the model materials have been quantified. Values of two additional constitutive properties, the characteristic length and the coupling number, which are present within micropolar elasticity but absent from its classic counterpart have also been determined. The dependence of these additional properties on void size has been investigated and characteristic length values compared to the length scales inherent within the structure of the model materials

Is smaller always stiffer? On size effects in supposedly generalized continua

Heterogeneous materials having constitutive behaviour described by more generalized continuum theories incorporating additional degrees of freedom such as couple stress, micropolar or micromorphic elasticity are expected to exhibit size effects in which there is an apparent increase in stiffness as the size scale reduces. Here we briefly demonstrate that for a simple heterogeneous material the size effect predicted when loaded in bending depends on the nature of the sample surface. Diverse size effects may thus be exhibited by the same material. We then show by detailed finite element analysis of a more representative material with regular heterogeneity that this diversity of size effects might actually be observed in practice thereby providing an explanation for the contradictory size effects that have sometimes been reported for real materials

NONLINEAR ELASTICITY EFFECTS IN THE DYNAMIC FRACTURE OF POLYMERS

Un modèle élastique non linéaire de l'essai de rupture en double torsion à grande vitesse (HSDT), utilisé pour tester des tuyaux de polyethylène, donne une dispersion des valeurs de la résistance à la rupture dynamique. Une technique permettant de déterminer le module de cisaillement approprié au HSDT a été développée. Les données obtenues montre que les modules de cisaillement chutent lorsque le cisaillement croît. Le modèle numérique a été modifié pour inclure ce comportement élastique non linéaire. Les valeurs de résistance ainsi recalculées sont beaucoup moins dispersées.A linear elastic model of the High Speed Double Torsion fracture test, which is used to test pipe grade polyethylenes, produces scattered dynamic fracture resistance values. A technique that determines shear modulus data appropriate to the HSDT has developed. The data obtained from this technique indicate that the shear moduli fall as the applied shear strain is increased. The numerical model has been modified to include this nonlinear elastic behaviour. The recomputed resistance data shows significantly less scatter

The micropolar elastic behaviour of model macroscopically heterogeneous materials

This paper describes the design, manufacture, testing and analysis of two model heterogeneous materials that exhibit non classical elastic behaviour when loaded. In particular both materials demonstrate a size effect in which stiffness increases as test sample size reduces; an effect that is unrecognized by classical elasticity but predicted by more generalized elasticity theories that are thought to describe the behaviour of heterogeneous materials more fully. The size effect has been observed by both experimental testing and finite element analysis that fully incorporates the details of the underlying heterogeneity designed into each material. The size effect has been quantified thus enabling both the modulus and also the characteristic length, an additional constitutive parameter present within micropolar and other generalized elasticity theories, to be determined for each material. These characteristic length values are extraordinarily similar to the length scales associated with the structure of the materials. An additional constitutive parameter present within plane micropolar elasticity theory that quantifies shear stress asymmetry has also been determined for one of the materials by using an iterative process that seeks to minimize the differences between numerical predictions and test results