Size effects exhibited by mechanical metamaterials when loaded may be positive such that reducing overall size towards that of the length scale of the underlying structure intrinsic to the material is accompanied by increasing stiffness or rigidity, a phenomenon that has been repeatedly observed and is also forecast by various more generalised continuum theories of deformation in loaded heterogeneous continua. However, such effects may in certain circumstances be contradictory in that decreasing size is accompanied by increasing compliance, the transition from the conventional, positive to this theoretically unanticipated negative behaviour having been explained recently in terms of the distribution of material within 2D continua subject to bending. Here we report on a novel phenomenon newly observed in periodic 3D lattice materials comprised of repeated cubic unit cells formed of exterior edge and interior diagonal connectors. Subtle redistribution of matrix material from edges to diagonals causes the size effect to change dramatically, inverting from positive to negative when loaded in the torsional mode while the corresponding effect for the flexural mode remains entirely positive under the same circumstances. This observation may lead to the prospect of optimising the design of 3D periodic metamaterials to provide a stiffer response in one loading mode and a more compliant response in another, a feature that could potentially be exploited in various innovative applications.

In this study, we developed a predictive model of in vivo stent based drug release and distribution that is capable of providing useful insights into performance. In a combined mathematical modelling and experimental approach, we created two novel sirolimus-eluting stent coatings with quite distinct doses and release kinetics. Using readily measurable in vitro data, we then generated parameterised mathematical models of drug release. These were then used to simulate in vivo drug uptake and retention. Finally, we validated our model predictions against data on drug kinetics and efficacy obtained in a small in vivo evaluation. In agreement with the in vivo experimental results, our mathematical model predicted consistently higher sirolimus content in tissue for the higher dose stents compared with the lower dose stents. High dose stents resulted in statistically significant improvements in three key efficacy measures, providing further evidence of a basic relationship between dose and efficacy within DES. However, our mathematical modelling suggests a more complex relationship is at play, with efficacy being dependent not only on delivering an initial dose of drug sufficient to achieve receptor saturation, but also on the consequent drug release rate being tuned to ensure prolonged saturation. In summary, we have demonstrated that our combined in vitro experimental and mathematical modelling framework may be used to predict in vivo DES performance, opening up the possibility of an in silico approach to optimising the drug release profile and ultimately the effectiveness of the device.

This paper reports on the influence that a periodic microstructure has on the unconstrained flexural vibration of geometrically similar but differently sized heterogeneous beam samples. A numerical investigation was conducted by finite element analysis (FEA) incorporating the detailed heterogeneity to identify and quantify any effect of beam size on the transverse modal frequencies when the microstructural scale is comparable to the overall size. Finite element models of the macroscopic beam samples were created by firstly specifying microstructural scale unit cells containing a single void or inclusion using ANSYS Mechanical APDL and then repeatedly regenerating these as required. Four beam sizes consisting of one, two, three or four layers of unit cells were created while the length to depth aspect ratio was kept constant for all sizes. Void or inclusion volume fraction was also altered while keeping the homogenised mass and stiffness properties of each beam fixed. The influence of the beam boundary texture on the results was also investigated. The ANSYS results were compared to the analytical solution for a conventional Timoshenko beam and a nonlocal Timoshenko beam. Using the nonlocal Timoshenko analysis, the Eringen small length scale coefficients were estimated but found to be size dependent. Numerical predictions obtained from a novel control volume based finite element (CVFEM) procedure incorporating micropolar constitutive behaviour were therefore matched to the ANSYS results and thereby used to identify the two additional constitutive parameters featuring in planar micropolar elasticity theory, namely the characteristic length in bending and coupling number.

A prominent size effect has previously been reported for the fracture behaviour of brittle porous materials, with smaller specimens behaving quite differently to their larger counterparts. In such materials, the size of the K-dominant zone has been numerically found to be greatly affected by the presence of voids in the near-tip area, thus putting the assumption of a single fracture parameter under question. In order to address this, in this study mode I tests are conducted on porous double cantilever beam specimens, while the stress distribution in the near-tip area is being observed by means of photoelasticity. Results validate the predicted size effect and suggest that the voids can indeed alter the size and shape of the stress pattern in the specimens. A parametric study is then conducted to investigate the influence of void shape variations that can be caused by manufacturing inaccuracies on the stress concentration at the crack tip. It is found that although the stress intensity at the crack tip can be greatly affected by such factors, the size of the K-dominant zone remains unaffected.

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.

In this paper the influence of microstructure on the free vibration of geometrically similar heterogeneous beams with free-free boundary conditions was numerically investigated by detailed finite element analysis (FEA) to identify and quantify any effect of beam size on transverse modal frequencies when the microstructural scale is comparable to the overall size. ANSYS Mechanical APDL was used to generate specific unit cells at the microstructural scale comprised of two isotropic materials with different material properties. Unit cell variants containing voids and inclusions were considered. At the macroscopic scale, four beam sizes consisting of one, two, three or four layers of defined unit cells were represented by repeatedly regenerating the unit cell as necessary. In all four beam sizes the aspect ratio was kept constant. Changes to the volume fractions of each material were introduced while keeping the homogenized properties of the beam fixed. The influence of the beam surface morphology on the results was also investigated. The ANSYS results were compared with the analytical results from solution to Timoshenko beam and nonlocal Timoshenko beam as well as numerical results for a Micropolar beam. In nonlocal Timoshenko beams the Eringen's small length scale coefficients were estimated for some of the studied models. Numerical analyses based on Micropolar theory were carried out to study the modal frequencies and a method was suggested to estimate characteristic length in bending and coupling number via transverse vibration which verifies the use of Micropolar elasticity theory in dynamic analysis.

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Linear Elastic Fracture Mechanics usually only considers the singular stresses when describing the conditions under which fracture would occur in a brittle material. However, it is becoming more widely recognised that non-singular stresses can become significant depending on the geometry and configuration of the specimen. This study investigates the impact of non-singular stresses on the stress intensity of low to medium porosity brittle materials. To address this, discrete finite element models of Double Cantilever Beam (DCB) samples were created and the full near-tip stress field in mode I loading was numerically evaluated. A parametric study was conducted, examining the influence of overall specimen size, material porosity and crack tip location relative to the nearest void. Results indicate a prominent size effect on the stress intensity at the crack tip of porous materials, with smaller specimen exhibiting tougher behaviour than their respective larger counterparts. This size effect, which is amplified with increasing porosity, is closely correlated with the variation of non-singular stresses, both parallel and normal to the crack plane. A model to predict the behaviour of porous specimen for different sizes is suggested based on the findings.

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.

The authors would like to draw attention to the fact that we have identified a numerical implementation error which affects some of the results and conclusions of the paper ''Does anisotropy promote spatial uniformity of stent-delivered drug distribution in arterial tissue?".Most significantly, the conclusion that the convection dominated case results in the highest levels of uniformity is replaced with the conclusion that the diffusion dominated case in fact results in the highest levels of uniformity, followed by the reaction dominated case and lastly by the convection dominated case.The updated plots for the convection dominated case are presented below (Fig. 1) whilst the plots for the diffusion and reaction dominated cases are not provided, since they are visibly very similar to those of the original paper.However, we provide updated numerical values (which quantify the uniformity and agreement between the one-dimensional and three-dimensional models) in Tables 1 and2 for all three cases (convection dominated, diffusion dominated, reaction dominated).While the newly tabulated results for the diffusion and reaction dominated cases show some change, these data represent differences between concentration profiles that have visibly not changed greatly.These data are thus only provided for the sake of completeness.Values referred to in the text of the original paper should be regarded as updated by the values of the tables presented here.The text of Sections 4.1.1 and 4.2.1 is replaced by the updated text below.The plots associated with Section 4.5 (varying strut thickness and separation) are also affected and we provide below in Fig. 2 the updated plots for the convection dominated case (diffusion dominated and reaction dominated figures are visibly very similar to the original paper).Finally, for the benefit of the reader, we provide a completely revised Section 5.

Composite laminates are usually assumed to be homogeneous when determining the energy release rates (ERRs) associated with inter-ply delamination. This short paper discusses the effect of neglecting this assumption by accounting for inter-ply interface layer thickness and the resulting influence that this may have on the ERRs. A global approach is used to analytically determine ERRs for delaminations subject to mixed mode loading in symmetric double cantilever beam (DCB) samples of a material formed of alternating stiff and compliant layers. In contrast to their homogeneously determined counterparts these ERRs and their mixity are dependent on both sample depth and interface thickness and when compared the conditions under which obvious differences become apparent can be explicitly identified. Some brief conclusions on the application of the analysis to the prescription of practical delamination testing protocols for composite laminates are drawn.

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.

Heterogeneous materials having constitutive behaviour described by more generalised 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.

In this article we investigate the role of anisotropic diffusion on the resulting arterial wall drug distribution following stent-based delivery. The arterial wall is known to exhibit anisotropic diffusive properties, yet many authors neglect this, and it is unclear what effect this simplification has on the resulting arterial wall drug concentrations. Firstly, we explore the justification for neglecting the curvature of the cylindrical arterial wall in favour of using a Cartesian coordinate system. We then proceed to consider three separate transport regimes (convection dominated, diffusion dominated, reaction dominated) based on the range of parameter values available in the literature. By comparing the results of a simple one-dimensional model with those of a fully three-dimensional numerical model, we demonstrate, perhaps surprisingly, that the anisotropic diffusion can promote the spatial uniformity of drug concentrations, and furthermore, that the simple analytical one-dimensional model is an excellent predictor of the three-dimensional numerical results. However, the level of uniformity and the time taken to reach a uniform concentration profile depends on the particular regime considered. Furthermore, the more uniform the profile, the better the agreement between the one-dimensional and three-dimensional models. We discuss the potential implications in clinical practice and in stent design. (C) 2015 The Authors. Published by Elsevier Ltd.

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.

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.

Drug-eluting stents have significantly improved the treatment of coronary artery disease. Compared with their bare metal predecessors, they offer reduced rates of restenosis and thus represent the current gold standard in percutaneous coronary interventions. Drug-eluting stents have been around for over a decade, and while progress is continually being made, they are not suitable in all patients and lesion types. Furthermore there are still real concerns over incomplete healing and late stent thrombosis. In this paper, some modelling approaches are reviewed and the future of modelling and simulation in this field is discussed.