On the nonlinear relationship between wall shear stress topology and multi-directionality in coronary atherosclerosis

Background and Objective: In this paper we investigate twelve multi-directional/topological wall shear stress (WSS) derived metrics and their relationships with the formation of coronary plaques in both computational fluid dynamics (CFD) and dynamic fluid-structure interaction (FSI) frameworks. While low WSS is one of the most established biomechanical markers associated with coronary atherosclerosis progression, alone it is limited. Multi-directional and topological WSS derived metrics have been shown to be important in atherosclerosis related mechanotransduction and near-wall transport processes. However, the relationships between these twelve WSS metrics and the influence of both FSI simulations and coronary dynamics is understudied. Methods: We first investigate the relationships between these twelve WSS derived metrics, stenosis percentage and lesion length through a parametric, transient CFD study. Secondly, we extend the parametric study to FSI, both with and without the addition of coronary dynamics, and assess their correlations. Finally, we present the case of a patient who underwent invasive coronary angiography and optical coherence tomography imaging at two time points 18 months apart. Associations between each of the twelve WSS derived metrics in CFD, static FSI and dynamic FSI simulations were assessed against areas of positive/negative vessel remodelling, and changes in plaque morphology. Results: 22–32% stenosis was the threshold beyond which adverse multi-directional/topological WSS results. Each metric produced a different relationship with changing stenoses and lesion length. Transient haemodynamics was impacted by coronary dynamics, with the topological shear variation index suppressed by up to 94%. These changes appear more critical at smaller stenosis levels, suggesting coronary dynamics could play a role in the earlier stages of atherosclerosis development. In the patient case, both dynamics and FSI vs CFD changes altered associations with measured changes in plaque morphology. An appendix of the linear fits between the various FSI- and CFD-based simulations is provided to assist in scaling CFD-based results to resemble the compliant walled characteristics of FSI more accurately. Conclusions: These results highlight the potential for coronary dynamics to alter multidirectional/topological WSS metrics which could impact associations with changes in coronary atherosclerosis over time. These results warrant further investigation in a wider range of morphological settings and longitudinal cohort studies in the future.Harry J. Carpenter, Mergen H. Ghayesha, Anthony C. Zander, Peter J. Psalti

Vibrations of axially travelling CNT reinforced beams with clamped-clamped boundary condition and an elastic support

Vibrational analysis in engineering systems of axially travelling beams has attracted noticeable attention due to the many applications, such as in robotic manipulators, cable tramways, textile fibres, and in general when there is the axial mass transport of a continuous structure. This article studies the vibrational response of axially-travelling, functionally-graded, carbon nanotube-(CNT)-reinforced beam structures, by investigating linear gyroscopic aspects, such as Argand diagrams. The distribution of CNT fibres is assumed to vary along the thickness of the beam. The Hamilton principle is employed to obtain the coupled axial and transverse behaviour of the beam, subjected to clamped-clamped boundary condition and additionally supported by a spring. These equations of motion are then solved using the modal decomposition technique for the Coriolis-dependent axial and transverse frequencies. For verification, the results are compared to the simplified case in the literature for CNT strengthened beams with zero axial velocity, the dynamics of axially travelling beams, studies of the clamped-clamped boundary condition, and the effects on the Argand diagrams, which have been performed. The Argand diagrams are plotted to examine the effects of varying axial speed on the different linear characteristics of vibration. Variation of the volume fraction of the CNT and the spring support, has also been considered, to understand its effects on the vibration characteristics. Results produced in this article are important in assisting in the future design of engineering devices involving axially travelling systems.Moaz Sibtain, Saxon Smith, Alireza Yeganehmehr, Oscar Zi Shao Ong, Mergen H. Ghayes

Extremely large oscillations of cantilevers subject to motion constraints

The nonlinear extremely large-amplitude oscillation of a cantilever subject to motion constraints is examined for the first time. In order to be able to model the large-amplitude oscillations accurately, the equation governing the cantilever centerline rotation is derived. This allows for analyzing motions of very large amplitude even when tip angle is larger than π/2. The Euler–Bernoulli beam theory is employed along with the centerline inextensibility assumption, which results in nonlinear inertial terms in the equation of motion. The motion constraint is modeled as a spring with a large stiffness coefficient. The presence of a gap between the motion constraint and the cantilever causes major difficulties in modeling and numerical simulations, and results in a nonsmooth resonance response. The final form of the equation of motion is discretized via the Galerkin technique, while keeping the trigonometric functions intact to ensure accurate results even at large-amplitude oscillations. Numerical simulations are conducted via a continuation technique, examining the effect of various system parameters. It is shown that the presence of the motion constraints widens the resonance frequency band effectively which is particularly important for energy harvesting applications.Hamed Farokhi, Mergen H. Ghayes

Nonsymmetric nonlinear dynamics of piezoelectrically actuated beams

The nonlinear behavior of a piezoelectrically actuated clamped–clamped beam has been examined numerically while highlighting the nonsymmetric response of the system. The nonlinearly coupled electromechanical model of the piezoelectric beam system is developed employing the Bernoulli–Euler theory along with the piezoelectric stress–voltage equations. A general nonsymmetric configuration is considered with a piezoelectric patch partially covering the beam. The geometric nonlinearities of stretching type are taken into account for both the piezoelectric patch and the beam. Through use of the generalized Hamilton's principle, the nonlinearly coupled electromechanical equations of transverse and longitudinal motions of the piezoelectrically actuated beam are derived. A high-dimensional Galerkin scheme is utilized to recast the equations of partial differential type into ordinary differential type. For comparison and benchmark purposes, a three-dimensional finite element model is developed using abaqus/cae to verify the model developed in this study. It is shown that the response of the system is strongly nonsymmetric and that it is essential to retain many degrees-of-freedom to ensure converged results.Mergen H. Ghayesh, Hamed Farokh

Nonlinear mechanics of beams with partial piezoelectric layers

This paper investigates the nonlinear static response as well as nonlinear forced dynamics of a clamped–clamped beam actuated by piezoelectric patches partially covering the beam from both sides. This study is the first to develop a high-dimensional nonlinear model for such a piezoelectric-beam configuration. The nonlinear dynamical resonance characteristics of the electromechanical system are examined under simultaneous DC and AC piezoelectric actuations, while highlighting the effects of modal energy transfer and internal resonances. A multiphysics coupled model of the beam-piezoelectric system is proposed based on the nonlinear beam theory of Bernoulli–Euler and the piezoelectric constitutive equations. The discretized model of the system is obtained with the help of the Galerkin weighted residual technique while retaining 32 degrees-of-freedom. Three-dimensional finite element analysis is conducted as well in the static regime to validate the developed model and numerical simulation. It is shown that the response of the system in the nonlinear resonant region is strongly affected by a three-to-one internal resonance.Hamed Farokhi, Mergen H. Ghayes

Modified couple stress theory in orthogonal curvilinear coordinates

The formulations for the modified couple stress theory (MCST) are consistently derived in general orthogonal curvilinear coordinate systems. In particular, the expressions for the rotation vector, higher-order strain, and stress tensors, i.e., the rotation gradient tensor and the deviatoric part of the symmetric couple stress tensor, and the classical strain and stress tensors are derived for an arbitrary orthogonal curvilinear coordinate system. Additionally, using the theory of surfaces, the formulations for the MCST are derived for general doubly curved coordinates, which are more convenient to use for shells of arbitrary curvature. The expressions for special cases, i.e., cylindrical and spherical shells, are obtained. The MCST expressions derived in this study are comprehensive and generally and can be used for consistent utilisation of the MCST in any orthogonal curvilinear coordinate system.Hamed Farokhi, Mergen H. Ghayes