Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving □iR\Box^i R, □iRμν\Box^i R_{\mu\nu} and □iRμνρσ\Box^i R_{\mu\nu\rho\sigma}

In this paper, we aim to perform a systematical investigation on the field equations and Noether potentials for the higher-order gravity theories endowed with Lagrangians depending on the metric and the Riemann curvature tensor, together with $i$th ($i=1,2,\cdot\cdot\cdot$) powers of the Beltrami-d'Alembertian operator $\Box$ acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian $\sqrt{-g}L_R(R,\Box R,\cdot\cdot\cdot,\Box^m R)$ through the direct variation of the Lagrangian and a method based upon the conserved current. Next the parallel analysis is extended to a more generic Lagrangian $\sqrt{-g}L_{\text{Ric}}(g^{\mu\nu}, R_{\mu\nu},\Box R_{\mu\nu}, \cdot\cdot\cdot,\Box^m R_{\mu\nu})$, as well as to the generalization of the Lagrangian $\sqrt{-g}L_{\text{Ric}}$, which depends on the metric $g^{\mu\nu}$, the Riemann tensor $R_{\mu\nu\rho\sigma}$ and $\Box^i R_{\mu\nu\rho\sigma}$s. Finally, all the results associated to the three types of Lagrangians are extended to the Lagrangian relying on an arbitrary tensor and the variables via $\Box^i$ acting on such a tensor. In particular, we take into consideration of equations of motion and Noether potentials for nonlocal gravity models. For Lagrangians involving the variables $\Box^i R$, $\Box^i R_{\mu\nu}$ and $\Box^i R_{\mu\nu\rho\sigma}$, our investigation provides their concrete Noether potentials and the field equations without the derivative of the Lagrangian density with respect to the metric. Besides, the Iyer-Wald potentials associated to these Lagrangians are also presented.Comment: 73 pages, No figure

The contact mechanics and occurrence of edge loading in modular metal-on-polyethylene total hip replacement during daily activities

The occurrence of edge loading in hip joint replacement has been associated with many factors such as prosthetic design, component malposition and activities of daily living. The present study aimed to quantify the occurrence of edge loading/contact at the articulating surface and to evaluate the effect of cup angles and edge loading on the contact mechanics of a modular metal-on-polyethylene (MoP) total hip replacement (THR) during different daily activities. A three-dimensional finite element model was developed based on a modular MoP bearing system. Different cup inclination and anteversion angles were modelled and six daily activities were considered. The results showed that edge loading was predicted during normal walking, ascending and descending stairs activities under steep cup inclination conditions (≥55°) while no edge loading was observed during standing up, sitting down and knee bending activities. The duration of edge loading increased with increased cup inclination angles and was affected by the cup anteversion angles. Edge loading caused elevated contact pressure at the articulating surface and substantially increased equivalent plastic strain of the polyethylene liner. The present study suggested that correct positioning the component to avoid edge loading that may occur during daily activities is important for MoP THR in clinical practice