A note on the kinematic boundary condition with historical perspective

SummaryDisplacement‐based seismic assessment of buildings containing unreinforced masonry (URM) walls requires as input, among others, estimates of the in‐plane drift capacity at the considered limit states. Current codes assess the drift capacity of URM walls by means of empirical models with most codes relating the drift capacity to the failure mode and wall slenderness. Comparisons with experimental results show that such relationships result in large scatter and usually do not provide satisfactory predictions. The objective of this paper is to determine trends in drift capacities of modern URM walls from 61 experimental tests and to investigate whether analytical models could lead to more reliable estimates of the displacement capacity than the currently used empirical models. A recently developed analytical model for the prediction of the ultimate drift capacity for both shear and flexure controlled URM walls is introduced and simplified into an equation that is suitable for code implementation. The approach follows the idea of plastic hinge models for reinforced concrete or steel structures. It explicitly considers the influence of crushing due to flexural or shear failure in URM walls and takes into account the effect of kinematic and static boundary conditions on the drift capacity. Finally, the performance of the analytical model is benchmarked against the test data and other empirical formulations. It shows that it yields significantly better estimates than empirical models in current codes. The paper concludes with an investigation of the sensitivity of the ultimate drift capacity to the wall geometry, static, and kinematic boundary conditions.

Reconciling different formulations of viscous water waves and their mass conservation

Turbulence structure near a wavy-sheared air-water interface was numerically investigated using a direct numerical simulation (DNS). The unsteady three-dimensional Navier-Stokes equations were numerically solved on both air and water sides for a wind-driven turbulent flow with a Reynolds number Re=6700, based on the maximum wave height and uniform velocity on the air side. Two boundary conditions [1] were used to determine the instantaneous shape of air-water interface. One was the dynamical boundary condition determined from the balance of the normal and tangential stresses acting on the wavy air-water interface, and the other was the kinematic boundary condition that describes the Lagrangian behavior of the air-water interface. The results show that the organized motion intermittently appears on the front side of the wavy interface in the air flow. The organized motion in the air flow gives high shear and pressure on the interface. The high shear and pressure acting on the wavy air-water interface induce the organized surface-renewal motion in the water flow beneath the air-water interface, which controls heat and mass transfer across the interface. The predictions well explain the previous measurements in the wind wave tank [2].

The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries

Chapter 4 - Conservation Laws

Rapidly rotating convection-driven dynamos are investigated under different kinematic and magnetic boundary conditions using direct numerical simulations. At a fixed rotation rate, represented by the Ekman number$Ek=5\times 10^{-7}$, the thermal forcing is varied from$2$to$20$times its value at the onset of convection ($\mathcal {R}=Ra/Ra_c=2\unicode{x2013}20$, where$Ra$is the Rayleigh number), keeping the fluid properties constant ($Pr=Pr_m=1$, where$Pr$and$Pr_m$are the thermal and magnetic Prandtl numbers). The statistical behaviour of the dynamos, including the force balance, energetics, and heat transport, depends on the boundary conditions that dictate both the boundary layer and the interior dynamics. At a fixed thermal forcing ($\mathcal {R}=3$), the structure and strength of the magnetic field produced by the dynamos, especially near the walls, depend on both velocity and magnetic boundary conditions. Though the leading-order force balance in the bulk remains geostrophic, the Lorentz force becomes comparable to the Coriolis force inside the thermal boundary layer with no-slip, perfectly conducting conditions. In this case, a term signifying the work done by the Lorentz force in the turbulent kinetic energy (TKE) equation is found to have some components that extract energy from the velocity field to produce the magnetic field, while some other components extract energy from the magnetic field to produce TKE. However, with no-slip, pseudo-vacuum conditions, all the components of the work done by the Lorentz force perform unidirectional energy transfer to produce magnetic energy from the kinetic energy of the fluid to sustain dynamo action. We find heat transfer enhancement in the rotating dynamo convection, as compared with non-magnetic rotating convection, with the peak enhancement lying in the range$\mathcal {R}=3\unicode{x2013}4$. For free-slip conditions, in the absence of an Ekman layer, the dynamo action may alter the heat transport significantly by suppressing the formation of large-scale vortices. However, the highest heat transfer enhancement is found at$\mathcal {R}=3$with no-slip, perfectly conducting walls, which can be attributed to a local magnetorelaxation of the rotational constraint due to enhanced Lorentz force inside the thermal boundary layer.

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

On the role of higher-order conditions in distortion gradient plasticity

Objective: Frontal crashes in which the vehicle has poor structural engagement, such as small-overlap and oblique crashes, account for a large number of fatalities. These crash modes are characterized by large intrusion and vehicle yaw rotation. Results from previous studies have shown mixed results regarding the importance and effects of these parameters. The aim of this study was to evaluate how vehicle yaw rotation, instrument panel intrusion, and the time history of the pulse angle influence chest injury outcomes.Method: This study was conducted using kinematic boundary conditions derived from physical crash tests, which were applied on a finite element simulation model of a vehicle interior including a finite element human model. By performing simulations with different levels of simplified boundary conditions and comparing the results to a simulation with a full set of boundary conditions, the influence of the simplifications was evaluated. The injury outcome measure compared between the simulations was the expected number of fractured ribs. The 3 simplifications simulated were (1) removal of vehicle yaw rotation, (2) removal of vehicle yaw rotation plus an assumption of a constant pulse angle between the x- and y-acceleration, and (3) removal of instrument panel intrusion.The kinematic boundary conditions were collected from 120 physical tests performed at the Insurance Institute of Highway Safety; 77 were small-overlap tests, and 43 were moderate overlap tests. For each test, the full set of boundary conditions plus the 3 simplifications were simulated. Thus, a total of 480 simulations were performed.Results: The yaw rotation influences occupant interaction with the frontal airbag. For the approximation without this kinematic boundary component, there was an average error in injury outcome of approximately 13% for the moderate overlap cases. Large instrument panel intrusion increases the risk of rib fracture in nearside small-overlap crashes. The mechanism underlying this increased fracture risk is a combination of increased airbag load and a more severe secondary impact to the side structure. Without the intrusion component, the injury risk was underestimated by 8% for the small-overlap crashes.Conclusion: The approximation with least error was version 2; that is, a model assuming a constant pulse angle, including instrument panel intrusion but no vehicle yaw rotation. This approximation simulates a sled test with a buck mounted at an oblique angle. The average error for this approximation was as low as 2–4%.

Heat and mass transfer during bubble growth in an alternating electric field

An semi-analytical solution method is presented for the buckling analysis of plates with a reinforced cutout under both uniform and non-uniform compression load- ing. The cutout is reinforced on both sides of the plate. The solution for buckling response is achieved in two subsequent steps: pre-buckling response from in-plane stress analysis and the critical buckling load and its mode from a buckling analysis that utilizes the pre- buckling stresses. The pre-buckling stresses are ob- tained based on the principle of minimum potential energy while automatically satisfying the equilibrium equations and compatibility condition. Hence, the strain energy of the plate is directly achieved by boundary integrals only. However, there exist material and thick- ness discontinuities between the unreinforced and rein- forced regions of the plate. Unlike the pre-buckling equilibrium equations, the buckling equations, which are obtained from the Treftz criterion, are highly com- plex, and the potential functions automatically satisfy- ing these equations do not exist. Hence, the Treftz cri- terion is applied with the assumed transverse displace- ment functions utilizing the full domain integration. In both pre-buckling and buckling analyses, the form of the assumed functions (real or complex) does not neces- sarily satisfy the kinematic boundary conditions. The kinematic boundary conditions are applied by introduc- ing elastic springs along the boundaries, and by enforcing the displacement field to satisfy kinematic boundary conditions through energy minimization.

This paper deals with the effects of variable chemical reaction and variable electric conductivity on free convection and mass transfer flow of a viscous, incompressible and electrically conducting fluid over an inclined stretching sheet with variable heat and mass fluxes under the influence of Dufour and Soret effects. The non-linear boundary layer equations with boundary conditions are transferred into a system of non-linear ordinary differential equations using an established similarity transformation. These non-linear and locally-similar ordinary differential equations are solved numerically by applying Nachtsheim–Swigert shooting iteration technique with sixth-order Runge–Kutta integration scheme. Comparison with previously published work is obtained and excellent agreement is found. The effects of various parameters on the dimensionless velocity, temperature and concentration profiles as well as the local skin-friction coefficient, heat and mass transfer rate from the stretching sheet to the surrounding fluid are presented graphically and in tabulated form for a hydrogen-air mixture. The numerical results showed that chemical reaction parameter K, order of reaction n, Dufour number Df , Soret number Sr and heat (or mass) flux parameter r play a crucial role in the solutions.

Numerical investigation of the dusty Williamson fluid with the dependency of time has been done in current disquisition. The flow of multiphase liquid/particle suspension saturating the medium is caused by stretching of porous surface. The influence of magnetic field and heat generation/absorption is observed. It is assumed that particle has a spherical shape and distributed uniformly in fluid matrix. The unsteady two-dimensional problems are modeled for both fluid and particle phase using conservation of mass, momentum and heat transfer. The finalized model generates the non-dimensioned parameters, namely Weissenberg number, unsteadiness parameter, magnetic parameter, heat generation/absorption parameter, Prandtl number, fluid particle interaction parameter, and mass concentration parameters. The numerical solution is obtained. Locality of skin friction and Nusselt number is deliberately focused to help of tables and graphs. While inferencing the current article it is clearly observed that increment of Williamson parameter, unsteadiness parameter, magnetic parameter, volume fraction parameter, and mass concentration parameter reduces the velocity profile of fluid and solid particles as well. And increment of Prandtl number, unsteadiness parameter, volume fraction parameter, and mass concentration parameter reduces the temperature profile of fluid and solid particles as well.

A static drilling riser model using free boundary conditions

Use of the Fick's law of diffusion is inappropriate in multi-component systems comprised of more than two species since it results in local violation of overall mass conservation. One approach that is often used to circumvent this problem is the dilute approximation, in which the conservation of the diluent is not enforced. It is generally believed that if the mass fraction of the diluent is "large" and it is a non-reacting species, the dilute approximation yields accurate results. In this study, the accuracy of the dilute approximation is quantitatively assessed by comparing its predictions against a rigorous multi-component diffusion formulation, derived from the Stefan-Maxwell equation. Both mass and heat flux errors are computed for a one-dimensional pure diffusion problem. It is found that even for quite dilute mixtures with mass fraction of the diluent exceeding 70%, the errors are ~15% for mass fluxes and as high as 35% for heat fluxes. In addition, the errors are found to be strongly dependent on the type of species in the mixture, implying that the validity of the dilute approximation cannot be judged a priori. Additional results are presented to show that in systems with combined heat and mass diffusion, net heat transfer can occur against the imposed temperature gradient. A Second Law analysis is presented and performed to corroborate these non-intuitive results.