Nonlinear Inertia Terms Research Articles

Modelling ductile damage in metals and alloys through Weyl condition exploiting local gauge symmetries

Local translational and scaling symmetries in space–time is exploited for modelling ductile damage in metals and alloys over wide ranges of strain rate and temperature. The invariant energy density corresponding to the ductile deformation is constructed through the gauge invariant curvature tensor by imposing the Weyl like condition. In contrast, the energetics of the plastic deformation is brought in through the gauge compensating field emerged due to local translation and attempted to explore the geometric interpretation of certain internal variables often used in classical viscoplasticity models. Invariance of the energy density under the local action of translation and scaling is preserved through minimally replaced space–time gauge covariant operators. Minimal replacement introduces two non-trivial gauge compensating fields pertaining to local translation and scaling. These are used to describe ductile damage, including plastic flow and micro-crack evolution in the material. A space–time pseudo-Riemannian metric is used to lay out the kinematics in a finite-deformation setting. Recognizing the available insights in classical theories of viscoplasticity, we also establish a correspondence of the gauge compensating field due to spatial translation with Kröner’s multiplicative decomposition of the deformation gradient. Thermodynamically consistent coupling between viscoplasticity and ductile damage is ensured through an appropriate degradation function. Non-ordinary state-based (NOSB) peridynamics (PD) discretization of the model is used for numerical implementation. We conduct simulations of uniaxial deformation to validate the model against available experimental evidence and to assess its predictive features. The model’s viability is tested in reproducing a few experimentally known facts, viz., strain rate locking in the stress–strain response, whose origin is traced to a nonlinear microscopic inertia term arising out of the space–time translation symmetry. Finally, we solved 2D and axisymmetric deformation problems for qualitatively validating the model’s viability. NOSB peridynamics axisymmetric formulation in finite deformation setup is also presented.

Effect of nonlinear inertia and Duffing-type nonlinearity on the dynamic behaviour of nonlinear parametrically excited cantilever beams with tip mass

The dynamic response of parametrically excited cantilever beams with a tip mass (PECBM) is investigated in this paper. Employing the Method of Multiple Scales (MMS) and the Method of Varying Amplitudes (MVA), the frequency-response equations are obtained and the stability of the response is analysed. MVA results show that, for frequency ranges around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing-type nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations. These results are supported by numerical simulations obtained by direct integration (DI) of the equation of motion, while the MMS fails to predict the critical frequency at which this phenomenon occurs. Furthermore, based on the MVA, a criterion for determining the hardening and softening characteristics of PECBM is developed, showing that both Duffing nonlinearity and nonlinear inertia terms contribute to determining this characteristic. To verify the results, experimental measurements of the response of a PECBM are presented, showing good agreement with analytical and numerical results. It is shown that the mass added at the cantilever tip can change the characteristics of the system's dynamics, enhancing the softening behaviour of the PECBM.

Nonlinear modeling of a misaligned rotor under non-stationary operating conditions

In this study, a novel model is developed to investigate the angular acceleration of the nonlinear misaligned rotor passing through the critical speeds under non-stationary operating conditions. The concerned rotor is modeled elastic with the effects of nonlinear inertia terms and flexible bearing. Nonlinear equations of motion are derived using the Lagrange method and solved with the finite element method. Effects of parameters such as shaft eccentricity, damping, and stiffness of the supports on the increase in the rotor angular speed passing through the critical speed are considered. The obtained results indicate that a small amount of eccentricity prevents exceeding the elastic critical speed of the rotor. According to the results, if the support parameters are not selected properly, the rotor will not even be able to pass its rigid critical speed.

Nonlinear dynamics of parametrically excited cantilever beams with a tip mass considering nonlinear inertia and Duffing-type nonlinearity

The response of a parametrically excited cantilever beam (PECB) with a tip mass is investigated in this paper. The paper is mainly focused on accurate prediction of the response of the system, in particular, its hardening and softening characteristics when linear damping is considered. First, the method of varying amplitudes (MVA) and the method of multiple scales (MMS) are employed. It is shown that both Duffing nonlinearity and nonlinear inertia terms govern the hardening or softening behaviour of a PECB. MVA results show that for frequencies around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations, and results are validated by numerical results obtained from direct integration (DI) of the equation of motion, while the MMS fails to predict this critical frequency. A criterion for determining the hardening and softening characteristics of PECBs is developed and presented using the MVA. To verify the results, experimental measurements for a PECB with a tip mass are presented, showing good agreement with analytical and numerical results. Furthermore, it is demonstrated that the mass added at the cantilever tip can change the system characteristics, enhancing the softening behaviour of the PECB. It is shown that, within the frequency range considered, increasing the value of the tip mass decreases the amplitude response of the system and broadens the frequency range in which a stable response can exist.

Three-dimensional dynamics of a cantilevered pipe conveying pulsating fluid

This paper investigates the stability and three-dimensional (3-D) nonlinear dynamics of a cantilevered pipe with an internal fluid having a harmonic component of flow velocity superposed on a constant mean value. The nonlinear equations of motion for an inextensible cantilevered pipe with the consideration of internal pulsating flow are presented. The nonlinear inertial terms in the governing equations are replaced by equivalent displacement and velocity terms by using a perturbation method. The partial differential equations are then transformed into a set of ordinary differential equations (ODEs) by using the Galerkin method. The instability regions of the subcritical and supercritical resonances of a linear system are determined via the Floquet theory. The effects of mean flow velocity and mass ratio are investigated. The resulting coupled nonlinear differential equations are numerically solved using a fourth-order Runge-Kutta integration scheme for the subcritical and supercritical flow velocities. The nonlinear dynamical responses are presented in the form of bifurcation diagrams, time histories, phase portraits, power spectral densities (PSDs) and Poincaré maps. Some interesting and sometimes unexpected results have been observed with different flow velocities. The analytical model is found to exhibit rich and variegated dynamical behaviors which include 2-D or 3-D periodic, quasiperiodic and chaotic motions. The convergence analysis of the number of truncating modes in the Galerkin approach is also conducted.

Motion limiting nonlinear dynamics of initially curved beams

An initially curved beam is considered and its motion is constrained using two elastic constraints; the corresponding non-smooth nonlinear transverse dynamics is investigated for the first time. A clamped-clamped beam with one axially movable end is modelled via Bernoulli-Euler beam theory together with the inextensibility condition, giving rise to nonlinear inertial terms along with nonlinear geometric terms. Furthermore, the damping is modelled via Kelvin-Voigt internal damping model. The proposed model is verified for linear and nonlinear behaviours via comparison to a finite element model. The impact between beam and constraints is incorporated via calculating its work contribution. The nonlinear equation of motion is derived while incorporating geometric, damping, inertial, and constraints nonlinearities. A series of spatial basis functions together with corresponding vibration modes are used as the proposed solution of the transverse displacement. A modal discretisation is performed via the weighted-residual method of Galerkin and the corresponding non-smooth terms are kept intact while conducting numerical integration. A numerical continuation technique is utilised to solve the resultant equations. The non-smooth response is obtained for various cases and the effects of several parameters are studied thoroughly.

Nonlinear oscillations of viscoelastic microcantilever beam based on modified strain gradient theory

A viscoelastic microcantilever beam is analytically analyzed based on the modified strain gradient theory. The Kelvin Voigt scheme is used to model the beam viscoelasticity. Applying Bernoulli-Euler inextensibility of the centerline condition via Hamilton’s principle, the nonlinear equation of motion and related boundary conditions are derived based on shortening effect theory and discretized by Galerkin method. Inner damping, nonlinear curvature effect, and nonlinear inertia terms are applied. The generalized derived formulation in this article, allows modeling of any nonlinearity combinations such as nonlinear terms arises due to inertia, damping, and stiffness, as well as modeling the size effect via considering modified coupled stress or modified strain gradient theories. First mode nonlinear frequency and time response of the viscoelastic microcantilever beam are analytically evaluated utilizing multiple time scale method and validated by numerical findings. Results indicate that the nonlinear terms have an appreciable effect on natural frequency and time response of a viscoelastic microcantilever. Furthermore, the investigation reveals that due to the size effects, natural frequency enhances drastically, especially when the thickness of the beam and the length scale parameter are comparable. Outcomes clarify the importance of size effects in analyzing of the mechanical behavior of small scale structures.

Fluid transport with suspended particles by means of dilating peristaltic waves

In the present study, transport of a Newtonian fluid containing uniformly distributed tiny particles moving in a cylindrical tube is investigated when the peristaltic wave amplitude dilates exponentially. Long wavelength approximation is used to get rid of non-linear inertia terms. It is observed that dilation of wave-amplitude during propagation enhances pressure but reduces it with rising concentration of particulate suspension. Pressure rises with the amplitude dilation parameter in case solid particles are suspended in the fluid. Pressure is also observed to increase in the distal part of the tube.

Three-Dimensional Flows in a Rotating Cylinder in the Presence of Turbulent Boundary Layers on End Disks

An analytical model for calculating a viscous incompressible fluid flow in a rotating cylinder with a braking lid and the formation of turbulent boundary layers on the end surfaces is presented. The analysis is made with account for all nonlinear inertial terms in the equations of motion, within the framework of Loitsyanskii's integral relations. The approximate velocity profiles in the boundary layers are preassigned in accordance with the empirical 1/7 law. The main flow is subdivided into an inviscid quasisolid core and a lateral layer, where almost the entire upward part of a circulatory flow is concentrated. The unknown angular velocity of the core and its radial boundary are evaluated from the balance between the moments of the friction forces acting on the main rotating flow and the continuity condition for the circulatory flow.

A mixed upwind/central WENO scheme for incompressible two-phase flows

In this paper, an accurate and efficient numerical scheme is developed to solve the Navier-Stokes equations for two-phase flow systems with the nonlinear inertial term in conservation form. The major difficulty of discretizing the nonlinear inertial term in conservation form in two-phase flows is rooted in the discontinuous nature of momentum across interfaces. To overcome this difficulty, we developed the mixed Upwind/Central WENO scheme so that no grid deformation or mapping, or explicit interface reconstruction is necessary. Our mixed Upwind/Central WENO scheme also removes the dependence on the interface tracking/capturing method so it is compatible with any popular interface tracking/capturing methods. To decouple the overall equations of the two-phase system, an efficient semi-implicit projection scheme is constructed, which shares the efficiency of pressure stabilization method while enforces divergence-free condition to machine precision, to benefit interface advection, reduction consistency as well as large density ratio. The accuracy and reduction consistency of this projection scheme are validated by numerical experiments. Several benchmark cases are performed to show the capability of the overall scheme handling realistic two-phase problems with multiple physical effects, complex interface evolution and/or large density ratio of the order of (ρwater/ρair). Both quantitative and qualitative comparisons have been made and very good agreements with analytical solutions, published numerical and/or experimental results are achieved.

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.

A novel characteristic variational multiscale FEM for incompressible natural convection problem with variable density

PurposeThe purpose of this paper is to propose a new method to solve the incompressible natural convection problem with variable density. The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods and to deal with high Rayleigh number for the natural convection problem.Design/methodology/approachThe paper introduces a novel characteristic variational multiscale (C-VMS) finite element method which combines advantages of both the characteristic and variational multiscale methods within a variational framework for solving the incompressible natural convection problem with variable density. The authors chose the conforming finite element pair (P2, P2, P1, P2) to approximate the density, velocity, pressure and temperature field.FindingsThe paper gives the stability analysis of the C-VMS method. Extensive two-dimensional/three-dimensional numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.Originality/valueExtensive 2D/3D numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.

A novel pressure-correction projection finite element method for incompressible natural convection problem with variable density

This work introduces a novel pressure-correction (PC) projection finite element method for solving the incompressible natural convection problem with variable density. The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods. We choose the conforming finite element pair to approximate the density, the velocity, the pressure, and the temperature field, respectively. Moreover, the stability analysis of the first-order PC projection finite element method is given. Finally, ample numerical results are presented to demonstrate the efficacy and accuracy of the proposed method.

Inertial and Darcy’s Terms Ratio in Boundary Layer at Fluid–Porous Medium Interface

The study considers the forced boundary-layer flow overlying the Darcy–Brinkman porous medium and gives a quantitative analysis of the nonlinear inertial terms in the Brinkman filtration equation. The inertial terms are shown to be larger than the Darcy’s drag near the porous medium interface. The applicability range of boundary-layer approach is determined. It is suitable in high-permeable media with moderate velocities of an external flow. If it is slow enough, the inertial terms can be omitted in spite of interface effect. On the other hand, fast external flow produces the filtration with large pore-scale Reynolds number; therefore, the Forchheimer’s drag should be taken into account. It is shown the Brinkman term as well as inertial terms have a significant role in boundary-layer formation within the porous medium.

The effect of non-linear inertia on dynamic response of asymmetric multi-story buildings

The goal of this paper is to inspect the influence of non-linear inertia on dynamic response of multi-story asymmetric buildings. In this study unlike conventional linear approach, the non-linear inertial terms are considered in the equations of motion. For considering non-linear inertia, the motion equations are derived in local rotary coordinates system. Also, the stiffness and damping parameters are defined in the rotary coordinates system. In the novel approach which is proposed in this paper, the motion equations are different with conventional linear approach. Numerical examples are presented to compare the response of the proposed model and conventional linear model. The modeled building response is derived under harmonic excitation. The results show that if non-linear inertia is considered, the dynamic response of asymmetric multi-story buildings may become different with conventional linear approach.

Adaptive control of rotationally non-linear asymmetric structures under seismic loads

This paper aims to inspect the effectiveness of the Simple Adaptive Control Method (SACM) to control the response of asymmetric buildings with rotationally non-linear behavior under seismic loads. SACM is a direct control method and was previously used to improve the performance of linear and non-linear structures. In most of these studies, the modeled structures were two-dimensional shear buildings. In reality, the building plans might be asymmetric, which cause the buildings to experience torsional motions under earthquake excitation. In this study, SACM is used to improve the performance of asymmetric buildings, and unlike conventional linear models, the non-linear inertial coupling terms are considered in the equations of motion. SACM performance is compared with the Linear Quadratic Regulator (LQR) algorithm. Moreover, the LQR algorithm is modified, so that it is appropriate for rotationally non-linear buildings. Active tuned mass dampers are used to improve the performance of the modeled buildings. The results show that SACM is successful in reducing the response of asymmetric buildings with rotationally non-linear behavior under earthquake excitation. Furthermore, the results of the SACM were very close to those of the LQR algorithm.

Numerical Solution of Conjugate Free Convection From a Vertical Fin Embedded in a Non-Darcy Porous Medium

Abstract The problem of conjugate free convection from a vertical fin embedded in a fluid-saturated porous medium is investigated. The governing nonlinear equations are solved iteratively by a highly implicit finite difference scheme. In this paper, the results based on four models, viz the Darcy model, the Brinkman model, the non-Darcian model with nonlinear inertia and viscous terms, and also the non-Darcian model with viscous, nonlinear inertia and velocity square terms, are compared. It is seen that fin cooling is more effective at higher Grashof or Darcy numbers due to stronger convection effects. The local Nusselt number is observed to increase with the Grashof or Darcy numbers and decrease slightly with the conduction–convection parameter. The limitation of the Darcy’s law is observed at higher values of permeability when the non-Darcian models are more relevant.

Effect of Inertial Terms on Fluid–Porous Medium Flow Coupling

The study considers an effect of the nonlinear inertial terms in the Brinkman filtration equation on the characteristics of coupled flows in a pure fluid and porous medium in the frameworks of two independent problems. The first problem is the forced boundary-layer flow overlying the Darcy–Brinkman porous medium. The Prandtl theory is used, and the self-similar equations are built to describe it. It is shown that the inertial terms have a valuable effect on the boundary-layer structure because of the large velocity gradient in the transition zone. The boundary-layer thickness in a porous medium rapidly grows at large Reynolds numbers. The velocity magnitude and gradient at the interface also change. The second independent problem is an analysis of the inertial terms effect on the flow stability. The neutral curves of the full and linearized flow models are built using the shooting method. They have different short-wave asymptotic, but there are no significant changes in the critical Reynolds numbers and corresponding wave numbers.

Circulation control in magnetohydrodynamic rotating flows

A model of a laminar viscous conducting flow, near a dielectric disc in a uniform magnetic field and in the presence of external rotation, is considered, where there is a uniform suction and an axial temperature gradient between the flow and the disc’s surface. It is assumed that the parameters of the suction or the magnetohydrodynamic (MHD) interaction are such that the nonlinear inertial terms, related to the circulation flow, are negligible in the differential equations of the MHD boundary layer on a rotating disc. Analysis of the motion and energy equations, taking the dependence of density on temperature into account, is carried out using the Dorodnitsyn transformation. The exact analytical solution for the boundary layer and heat transfer equations is obtained and analysed, neglecting the viscous and Joule dissipation. The dependence of the flow characteristics in the boundary layer on the rate of suction and the magnetic field induction is studied. It is shown that the direction of the radial flow in the boundary layer on a disc can be changed, not only by variation of the ratio between the angular velocities in the external flow and the boundary layer, but also by changing the ratio of the temperatures in these two flows, as well as by varying the hydrodynamic Prandtl number. The approximate calculation of a three-dimensional flow in a rotating cylinder with a braking disc (or lid) is carried out, demonstrating that a magnetic field slows the circulation velocity in a rotating cylinder.

Size-dependent large-amplitude oscillations of microcantilevers

The size-dependent oscillations of a microcantilever with a tip (end) mass and a spring support undergoing a large-amplitude motion is analysed theoretically, taking into account curvature-related nonlinearities. Modelling small-size effects via use of the modified couple stress theory, the size-dependent potential and kinetic energies of the system are obtained. The continuous models for the motion behaviour of the microcantilever are developed via use of an energy method on the basis of Hamilton’s principle. Application of the centreline-inextensibility in oscillation course of the microcantilever results in a continuous model of the system with nonlinear inertial terms, which when coupled with curvature nonlinearities produces a highly nonlinear system. A weighted-residual method is then employed to truncate the continuous model, yielding the reduced-order model of the microcantilever motion with a generalised-coordinate-dependent mass matrix (due to inertial nonlinearities); a coupled continuation-time-integration method is then employed for the numerical simulations. The large-amplitude oscillation behaviour of the system is examined by constructing the frequency–responses and force-responses. The effect of the size of the end-mass on the nonlinear oscillation behaviour of the microcantilever is analysed. The importance of taking into account different nonlinearity sources is discussed. It is shown that the modified couple stress theory results in a stronger softening behaviour when compared to the classical continuum mechanics.