Marcus Wheel

A finite volume method for solid mechanics incorporating rotational degrees of freedom

A novel finite volume (FV) based discretization method for determining displacement, strain and stress distributions in loaded two dimensional structures with complex geometries is presented. The method incorporates rotation variables in addition to the displacement degrees of freedom employed in earlier FV based structural analysis procedures and conventional displacement based finite element (FE) formulations. The method is used to predict the displacement fields in a number of test cases for which solutions are already available. The effect of mesh refinement upon the accuracy of the solutions predicted by the method is assessed. The results of this assessment indicate that the new method is more accurate than previous FV procedures incorporating displacement variables only, particularly in cases where bending is the predominant mode of deformation, and therefore the new method represents a significant advance in the development of this type of discretization procedure. Interestingly, the results of the accuracy assessment exercise also indicate that the new FV procedure is also more accurate than the equivalent FE formulation incorporating displacement and rotational degrees of freedom.

Release mechanism and parameter estimation in drug-eluting stent systems : analytical solutions of drug release and tissue transport

Drug-eluting stents have significantly improved the treatment of coronary artery disease. They offer reduced rates of restenosis compared with their bare-metal predecessors and are the current gold standard in percutaneous coronary interventions. Drug-eluting stents have been approved for use in humans since 2002 and yet, despite the intensive research activity over the past decade, the drug release mechanism(s) and the uptake into the arterial wall are still poorly understood. While stent manufacturers have focussed primarily on empirical methods, several mathematical models have appeared in the literature considering the release problem, the uptake problem and also the coupled problem. However, two significant challenges that remain are in understanding the drug release mechanism(s) and also the determination of the various parameters characterizing the system. These include drug diffusion coefficients and dissolution constants in the stent polymer coating as well as drug diffusion coefficients, binding/uptake rates and the magnitude of the transmural convection in the arterial wall. In this paper we attempt to address these problems. We provide analytical solutions which, when compared with appropriate experiments, may allow the various parameters of the system to be estimated via the inverse problem. The analytical solutions which we provide here for drug release in vitro may thus be used as a tool for providing insights into the mechanism(s) of release.

On the coupling number and characteristic length of micropolar media of differing topology

In planar micropolar elasticity theory, the degree of micropolarity exhibited by a loaded heterogeneous material is quantified by a dimensionless constitutive parameter, the coupling number. Theoretical predictions of this parameter derived by considering the mechanical behaviour of regular, two-dimensional lattices with straight connectors suggest that its value is dependent on the connectivity or topology of the lattice with the coupling number in a square lattice predicted to be notably higher than in its hexagonal counterpart. A second constitutive parameter reflecting the intrinsic lattice size scale, the characteristic length, is also predicted to be topology-dependent. In this paper, we compare the behaviour of alternative two-dimensional heterogeneous materials in the context of micropolar elasticity. These materials consist of periodic arrays of circular voids within a polymeric matrix rather than a lattice of straight connectors. Two material variants that differ only in their matrix topology are investigated in particular. Values of the additional micropolar constitutive parameters are obtained for each material from both experimental tests and finite-element analyses. The values determined for these parameters, particularly the coupling number, suggest that their topological dependence differs appreciably from the theoretical predictions of the lattice models.

Computational analysis of the size effects displayed in beams with lattice microstructures

The mechanical behaviour of finite element based computational representations of heterogeneous materials with regular or periodic cellular microstructure is compared to existing closed form analytical predictions of their constitutive behaviour available within the open literature. During the computational investigation, slender, geometrically similar rectangular beams of different sizes which are comprised of regular, repeating arrangements of square cellular microstructures were represented using the finite element analysis (FEA) software ANSYS. Flexural loading of the virtual samples reveals that the materials exhibit the theoretically forecast size effect from which the relevant material constitutive properties, notably the flexural modulus and characteristic length can be identified. Initial findings suggest that while there is agreement between the numerically determined and theoretically predicted moduli the characteristic lengths in bending, \(l_b\), calculated from the numerical data appear to differ from the theoretical forecasts. Moreover, the computational representations indicate that finite sized material samples are capable of exhibiting size effects not predicted by the more general higher order constitutive theories. Results indicate that the nature of the size effect appears to depend on the prescription of the sample surfaces with respect to the specified microstructure of the material. While these unanticipated size effects show qualitative agreement with that forecast for a simple laminate material comprised of alternating stiff and compliant layers the consequences may be profound for experimental mechanical testing of such materials.

A numerical investigation and experimental verification of size effects in loaded bovine cortical bone

In this paper, we present 2- and 3-dimensional finite element-based numerical models of loaded bovine cortical bone that explicitly incorporate the dominant microstructural feature: the vascular channel or Haversian canal system. The finite element models along with the representation of the microstructure within them are relatively simple: 2-dimensional models, consisting of a structured mesh of linear elastic planar elements punctuated by a periodic distribution of circular voids, are used to represent beam samples of cortical bone in which the channels are orientated perpendicular to the sample major axis, while 3-dimensional models, using a corresponding mesh of equivalent solid elements, represent those samples in which the canals are aligned with the axis. However, these models are exploited in an entirely novel approach involving the representation of material samples of different sizes and surface morphology. The numerical results obtained for the virtual material samples when loaded in bending indicate that they exhibit size effects not forecast by either classical (Cauchy) or more generalized elasticity theories. However, these effects are qualitatively consistent with those that we observed in a series of carefully conducted experiments involving the flexural testing of bone samples of different sizes. Encouraged by this qualitative agreement, we have identified appropriate model parameters, primarily void volume fraction but also void separation and matrix modulus by matching the computed size effects to those we observed experimentally. Interestingly, the parameter choices that provide the most suitable match of these effects broadly concur with those we actually observed in cortical bone.

Paradoxical size effects in composite laminates and other heterogeneous materials

Size effects in which there is an apparent increase in stiffness with reducing size scale are forecast in those heterogeneous materials that have constitutive behaviour described by more generalized continuum theories such as couple stress, micropolar or micromorphic elasticity. This short paper considers possibly the simplest heterogeneous material exhibiting such size effects, a two phase composite laminate consisting of alternating layers of stiff and compliant material, and shows that when loaded in bending the nature of the size effect actually depends on the composition of the sample surfaces. The laminate material is apparently capable of exhibiting a diversity of size effects some of which are compatible with the predictions of generalized continuum theories while others are contradictory. Another heterogeneous material consisting of a periodic or regular array of voids within a classically elastic matrix is then considered. Detailed finite element analysis shows that the diversity of size effects encountered in the laminate material may also be observed in this more representative material thereby providing some insight into the contradictory size effects that have sometimes been reported elsewhere in the literature.

Modelling micropolar behaviour in cortical bone

Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright

Computational Modelling and Experimental Characterisation of Heterogeneous Materials

Heterogeneous materials can exhibit behaviour under load that cannot be described by classical continuum elasticity. Beams in bending can show a relative stiffening as the beam depth tends to zero, a size effect. Size effects are recognised in higher order continuum elastic theories such as micropolar elasticity. The drawback of higher order theories is the requirement of additional constitutive relations and associated properties that are often difficult to establish experimentally. The determination of additional constitutive properties and the computational modelling of micropolar elasticity will be discussed in the context of a model heterogeneous material loaded in simple 3 point bending. The model material was created by drilling holes in aluminium bar in a regular pattern, with the hole axis normal to the plane of bending. The bending tests show that a size effect is present. These results are compared against a model of the detailed beam geometries in the finite element package ANSYS. Both the experimental and detailed FEA results are used to extract the additional micropolar elastic material properties. A comparison is then made against analytical solutions and numerical solutions using both a beam finite element and a 2D control volume method that each incorporate micropolar behaviour.

Mechanical testing of wood-glass composite mast sections

Glass fibre-sheathed wood cored composite materials may offer a lightweight, stiff alternative to carbon fibre for manufacturing masts for sailing dinghies. However, at present, very little data quantifying the mechanical response of these materials when loaded is available. Before manufacturing a prototype mast from these materials, it is necessary to acquire such data and determine how the various fabrication parameters influence the behaviour of these materials when loaded. To address this need, a series of wood-glass composite tubular structures representative of typical mast sections were constructed and loaded in a suitable testing machine. This paper reports the results of these tests and discusses how the parameters investigated influence the stiffness and strength of the structures. The results provide useful data on which to base the design of prototype masts.