Quasi-static Models Research Articles

A hidden convexity in continuum mechanics, with application to classical, continuous-time, rate-(in)dependent plasticity

A methodology for defining variational principles for a class of PDE (partial differential equations) models from continuum mechanics is demonstrated, and some of its features are explored. The scheme is applied to quasi-static and dynamic models of rate-independent and rate-dependent, single-crystal plasticity at finite deformation.

Generalized Quasi-Static Mooring System Modeling with Analytic Jacobians

This paper presents a generalized and efficient method for quasi-static analysis of mooring systems, including complex scenarios such as when shared mooring lines interconnect multiple floating wind or wave energy devices. While quasi-static mooring models are well established, most published formulations are focused on specific applications, and no publicly available implementations provide efficient handling of large mooring system networks. The present formulation addresses these gaps by: (1) formulating solutions for edge cases not typically supported by quasi-static models; (2) creating a fully generalized model structure such that any combination of mooring lines, point masses, and floating bodies can be assembled; and (3) deriving analytic expressions for the system Jacobians (stiffness matrices) so that systems with many degrees of freedom can be solved efficiently. These techniques form the theory basis of MoorPy, an open-source mooring analysis library. The model is demonstrated on nine scenarios of increasing complexity with features of interest for offshore renewable energy applications. When compared with steady-state results from a lumped-mass dynamic model, the results show that the quasi-static formulation accurately calculates profiles and tensions and that its analytic approach provides more efficient and reliable computation of system stiffness matrices than finite-differencing methods. These results verify the accuracy of the MoorPy model.

Orbit-attitude-structure-thermal coupled modelling method for large space structures in unified meshes

Large space structures would exhibit orbit-attitude-structure-thermal coupled effects in complicated space environment. Thus, a new thermal-structure coupled modelling method is proposed using gradient-deficient absolute nodal coordinate formulation beam elements. Compared to the previous methods, both axial and circumferential heat conduction are considered. The cubic interpolation in temperature field instead of the linear interpolation is adopted to obtain a unified-mesh temperature-displacement description. The gravitational force and gravity gradient are modelled, and the effects of attitude motion and structural deformation on solar radiation intensity are considered. In addition, three kinds of quasi-static thermal-structure coupled models are proposed based on Euler-Bernoulli beam theory. Based on the quasi-static models, the theoretical formulas are derived to effectively predict the thermally induced vibrations of the beam in space. Four numerical examples are studied to validate the proposed models, including the heat conduction examples with three boundary conditions and a thermal-structure coupled cantilever beam example. Based on the proposed formulation, the orbit-attitude-structure-thermal coupled dynamics of a multibody system consisted of a rigid body and a large flexible appendage is investigated considering the gravity gradient, solar radiation, Earth's shadow, and self-shadow.

Adaptive impact analysis in flexible multibody systems based on hierarchically refined IGA models

Usually, detailed impact simulation models within flexible multibody systems have to be set up manually rather than being generated automatically. This is because the process requires prior knowledge of the time and location of the impact, as well as the element resolution within the contact area. If the penalty method is used to determine the occurring contact forces, the corresponding penalty factor also needs to be determined manually. This work, however, presents an adaptive algorithm to simulate impacts within flexible multibody systems fully automatically using reduced isogeometric analysis models, the floating frame of reference formulation, and quasistatic contact models for an efficient but still accurate simulation. The adaptive algorithm detects impacts in the system, determines the contact locations on the bodies, refines the contact area, and determines the penalty factor, and therefore automatically simulates impacts. The work shows how to automatically simulate impacts in flexible multibody systems without user action or prior knowledge of impact location and size. The first application example simulates significant elastodynamic effects within a long flexible rod. The goal is to validate the algorithm by preserving the wave propagation and energy of the system. The second application example simulates the impacts of two flexible double pendulums. This setup is a suitable benchmark for the complete adaptive impact analysis procedure as the flexible double pendulums undergo large rigid body motions.

A semi-analytical time-domain model with explicit fluid force expressions for fluidelastic vibration of a tube array in crossflow

It is widely acknowledged that fluidelastic instability (FEI), among other mechanisms, is of the greatest concern in the flow-induced vibration (FIV) of tube bundles in steam generators and heat exchangers. A range of theoretical models have been developed for FEI analysis, and, in addition to the earliest semi-empirical Connors' model, the unsteady model, the quasi-steady model and the semi-analytical model are believed to be three advanced models predominant in the literature. The unsteady and the quasi-static models share the merits of having explicit fluid force expressions and ease of being implemented but require more experimental inputs, whereas the semi-analytical model requires fewer parameters due to its analytical nature but is hard, if not prohibitive, to derive explicit fluid force expressions. Since the fluid force formulations set in the core of development of FEI models, the understanding and in particular the implementation of the semi-analytical model has been impaired by the nonexistence of explicit fluid force expression. This issue becomes more profound in time-domain analysis whereby the simple harmonic assumption is discarded. Here we report a new semi-analytical time-domain (SATD) FEI model with explicit fluid force expressions. The new model allows a consistent frequency-domain stability analysis and more importantly a truly time-domain response analysis. The theory was validated by calculating linear stability thresholds of two typical tube array patterns and comparing against reported experimental data. We then present a nonlinear time-domain analysis of a single loosely-supported tube with piece-wise linear stiffness. The nonlinear and nonsmooth dynamics was probed in details by utilizing various techniques, playing an emphasis on characterizing and distinguishing the chaotic vibration. We found that the system follows a quasi-periodic route to chaos. Such an in-depth study of the nonlinear dynamics of tubes in crossflow has never been reported in the context of SATD model. Our results enrich the theory and provide a different approach for linear and nonlinear dynamics of tube bundles, which are essential for the subsequent fretting wear analysis.

In-situ rock shattering and strain localization along a seismogenic fault in dolostones (Monte Marine fault, Italian Central Apennines)

In-situ shattered rocks are often associated with seismogenic fault zones, but their mechanism of formation is still matter of debate, partly because of the limited number of field studies. Here we describe the characteristics of in-situ shattered rocks distribution along the NW-SE-striking seismogenic Monte Marine Fault (MMF) in the Italian Central Apennines. In the studied area, the MMF cuts through Mesozoic carbonates, is exhumed from <3 km depth and consists of two >5 km-long major hard-linked segments with normal kinematics. The linkage between the two fault segments occurs along a ∼2 km-long step-over zone with E-W trending faults and oblique-slip kinematics. To the northwest, fault-related shear deformation is localized in a ∼5 m-thick cataclastic fault core and off-fault deformation is dominated by in-situ shattered rocks up to ∼40 m-thick. Instead, in the step-over zone to the southeast, the in-situ shattered rocks are up to ∼500 m thick, particularly where MMF crosscuts older low-angle thrust faults.We integrated detailed field structural surveys with microstructural and grain size distribution analyses of the fault rocks to assess the mechanism of (1) formation of in-situ shattered rocks and, (2) progressive localization of shear deformation along the MMF. The obtained results, after the viability of several formation mechanisms (mechanical models) have been reviewed, support the hypothesis that the formation of in-situ shattered rocks was associated with the propagation of (multiple) seismic ruptures (mainshocks and aftershock sequences) within a mechanically heterogeneous fault zone. Heterogeneity is due to the occurrence of preexisting damage related to previous earthquakes, but also inherited from the older low-angle thrust faults. Therefore, we suggest that the origin of these shattered rocks is more compatible with seismic related processes than only with quasi-static fault growth models. On the other hand, the cataclastic fault core derived from the progressive accommodation of shear deformation within the in-situ shattered rock volumes during several seismic cycles. We conclude that the large volumes of in-situ shattered rocks are the result of seismic-related dissipative processes in a geometrically and mechanically heterogeneous fault zone. In this scenario, large volumes of in-situ shattered rocks are compliant low velocity zones which can influence the propagation of earthquake ruptures.

A general kinematic model for lubricated ball bearings based on the minimum energy hypothesis

Accurately predicting the performance of lubricated rolling elements is critical for certain key bearing applications. A widely employed simulation technique is quasi-static models, which although computationally efficient to implement, lack accuracy in predicting the kinematics of bearing components. This problem arises from their construction under kinematic assumptions, instead of solving the kinematic based on real frictional interactions. This paper presents a quasi-static model based on the minimum energy hypothesis for lubricated contacts, which determines the kinematic of the ball considering the effects of the frictional interactions governed by the elastohydrodynamic lubrication mechanism. The influence of kinematic variables on the sliding patterns and power loss distribution of the contact are analysed. The results are compared with other published quasi-static models, revealing substantial differences. The influence of the kinematic hypotheses on power losses are also studied at a bearing component scale. Finally, the effect of the governing lubrication mechanism of the contact is discussed, concluding that the limiting shear stress of the lubricant and the centrifugal forces have a significant effect on the kinematic of the ball.

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure

On the (lack of) representativeness of quasi-static variational fracture models for unstable crack propagation

The present work is devoted to prove that unstable crack propagation events do not comply with quasi-static hypotheses and thus should be modelled by dynamic approaches. Comprehensive supporting evidence is provided on the basis of three different analyses conducted on multi-ligament unstable fracture conditions, including a simplified Spring-Mass model, detailed quasi-static and dynamic Phase Field fracture models, and bespoke experiments with 3D printed specimens. The obtained results unequivocally show that neglecting the inertial effects can lead to unsafe predictions in the presence of energetic barriers for the development of fracture. Likewise, quasi-static Phase Field fracture models are proven to yield crack patterns that disagree with the experimental evidence because they overlook the progressive diffusion of the mechanical information within the continuum. Moreover, the inability of quasi-static approaches to follow unstable crack propagation is shown to weaken the crucial irreversibility condition of fracture. Overall, these experimentally backed insights should be gravely reckoned with, for they are not exclusive to Phase Field fracture models but common to (almost) any variational approach to fracture, inter alia Cohesive Zone Models or Continuum Damage Mechanics.

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes

Abstract We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary Γ-convergence.

Validating mechanistic models of fluid displacement during imbibition

Pore-scale modelling is an important tool to improve our understanding of multiphase flow in porous media. Slow fluid invasion is commonly modelled using quasi-static pore network models (PNM). These models simulate the invasion in a network of simplified pores and throats by invading network elements in order of the quasi-static “invasion” capillary pressure needed for the invading fluid to enter. Despite a multitude of studies that address the predictiveness of PNM, it remains unclear what the leading causes of errors in these methods are, particularly during imbibition. To address this, we developed a novel method to directly validate the invasion capillary pressure models that underlie quasi-static PNM for the first time. The new method compares these models to local capillary pressures measured during in-situ flow experiments visualized with 4D μCT. We applied this to two different open-source PNM extractions from a μCT dataset of a glass beads pack that underwent slow imbibition. This methodology is limited by the temporal resolution of the data, hence we tested assumptions regarding displacement sequences when individual displacements could not be resolved. To constrain the uncertainty on the input parameters, we used local contact angles measured from the μCT images. In the PNMs we investigated, the model-predicted invasion-Pc values were on average greater than the direct measurements made using curvatures from the μCT images. Important sources of mismatch were the difficulty to accurately describe the pore space of a beadpack as a network of pores and throats, as well as the relatively low temporal resolution of the μCT dataset. The method presented here can be used to direct the development of improved pore network models.

Modeling multi-contact point physical interaction between the anthropomorphic finger and soft robotic exo-digit for wearable rehabilitation robotics applications.

Introduction: Effective control of rehabilitation robots requires considering the distributed and multi-contact point physical human-robot interaction and users' biomechanical variation. This paper presents a quasi-static model for the motion of a soft robotic exo-digit while physically interacting with an anthropomorphic finger model for physical therapy. Methods: Quasi-static analytical models were developed for modeling the motion of the soft robot, the anthropomorphic finger, and their coupled physical interaction. An intertwining of kinematics and quasi-static motion was studied to model the distributed (multiple contact points) interaction between the robot and a human finger model. The anthropomorphic finger was modeled as an articulated multi-rigid body structure with multi-contact point interaction. The soft robot was modeled as an articulated hybrid soft-and-rigid model with a constant bending curvature and a constant length for each soft segment. A hyperelastic constitute model based on Yeoh's 3rdorder material model was used for modeling the soft elastomer. The developed models were experimentally evaluated for 1) free motion of individual soft actuators and 2) constrained motion of the soft robotic exo-digit and anthropomorphic finger model. Results and Discussion: Simulation and experimental results were compared for performance evaluations. The theoretical and experimental results were in agreement for free motion, and the deviation from the constrained motion was in the range of the experimental errors. The outcomes also provided an insight into the importance of considering lengthening for the soft actuators.

AN IMPROVED ENGINEERING MODEL FOR CALCULATING THE RADIAL STIFFNESS OF ROLLER BEARINGS

The paper presents an analysis of quasi-static models for calculating stiffness and substantiates the choice of the most effective engineering model for calculating the radial stiffness of roller bearings. The analysis of quasi-static models for calculating the stiffness of roller bearings is carried out, which consists in numerical-analytical modeling of force-displacement ratios under conditions of contact interaction. It is noted that for roller bearings, semi-empirical dependencies proposed by Palmgren, Jones and Harris are more often used. For all numerical-analytical quasistatic models of roller bearings, a generalizing relationship between force and displacement at contact is introduced within the framework of the Hertz contact theory, where the stiffness characteristics are determined by the solution of the corresponding contact problem. The numerical values of constants obtained by several authors on the basis of empirical approaches or an approximate solution of the contact Hertz problem for steel bearings are given. In this paper, an analysis of quasi-static models for calculating stiffness is presented and the choice of the most effective engineering model for calculating the radial stiffness constants of roller bearings is substantiated. To solve the contact problem of a separate rolling element and rolling tracks, a finite element periodic model consisting of two halves of the rolling element is used, which are in contact separately with part of the inner and outer bearing rings, respectively, and have physically justified boundary conditions. The labor and computer time required to implement the proposed approach is less than when using the finite element method to determine the stiffness "in the forehead" for each individual bearing, which makes this model an effective engineering model for calculating the stiffness of roller bearings.

Quasi-Static Modelling of a Full-Channel Effective Magnetorheological Damper with Trapezoidal Magnetic Rings.

Magnetorheological damper (MRD) has been successfully applied to vehicle suspension systems as an intelligent core component. Most conventional MRDs have closed rectangle-shaped magnetic circuits, resulting in a short effective working length and negligible damping force. To address the above issues, a novel full-channel effective MRD with trapezoidal magnetic rings (FEMRD_TMR) is proposed. The trapezoidal magnetic ring can shunt the magnetic circuit, distributing it evenly along the damping channel and increasing the effective working length. Additionally, which has the same variation trend as the magnetic flux through it, makes the magnetic induction intensity distribution more uniform to reduce the magnetic saturation problem. Theoretically analyzing the damping characteristics of the FEMRD_TMR, a quasi-static model is developed to forecast the output damping force. The structural design of MRD is challenging since conventional quasi-static models rely on the yield stress of magnetorheological fluid (MRF) to reflect the rheological property, which cannot be directly observed and is challenging to calculate. The Takagi-Sugeno (T-S) fuzzy neural network and a unique magnetic circuit computation are offered as a novel quasi-static modeling approach to address the issue. The MRF's yield stress is linearized into magnetic induction intensity functions by the T-S fuzzy neural network and then converted into the MRD's structural size by the special magnetic circuit calculation. Therefore, the proposed quasi-static model can directly reflect the relationship between the damping force and structure size, simplifying MRD's structure design. The novel quasi-static model is shown to be more straightforward and understandable than the conventional Bingham quasi-static model and to have approximately accurate damping force prediction when compared to experimental data.

Turbulence dynamics and flow speeds in the inner solar corona: results from radio-sounding experiments by the Akatsuki spacecraft

ABSTRACT The solar inner corona is a region that plays a critical role in energizing the solar wind and propelling it to supersonic and supra-Alfvénic velocities. Despite its importance, this region remains poorly understood because of being least explored due to observational limitations. The coronal radio-sounding technique in this context becomes useful as it helps in providing information in parts of this least explored region. To shed light on the dynamics of the solar wind in the inner corona, we conducted a study using data obtained from coronal radio-sounding experiments carried out by the Akatsuki spacecraft during the 2021 Venus-solar conjunction event. By analysing X-band radio signals recorded at two ground stations (Indian Deep Space Network in Bangalore and Usuda Deep Space Center in Japan), we investigated plasma turbulence characteristics and estimated flow speed measurements based on isotropic quasi-static turbulence models. Our analysis revealed that the speed of the solar wind in the inner corona (at heliocentric distances from 5 to 13 solar radii), ranging from 220 to 550 km s−1, was higher than the expected average flow speeds in this region. By integrating our radio-sounding results with extreme ultraviolet (EUV) images of the solar disc, we gained a unique perspective on the properties and energization of high-velocity plasma streams originating from coronal holes. We tracked the evolution of fast solar wind streams emanating from an extended coronal hole as they propagated to increasing heliocentric distances. Our study provides unique insights into the least-explored inner coronal region by corroborating radio-sounding results with EUV observations of the corona.

Dynamic characteristics of spindle-bearing with tilted pedestal and clearance fit

Spindle-bearing systems are widely used in machine tools, aero engines, and other industrial equipment. The vibration characteristics of the spindle system may be affected by the pedestal tilt due to assembly errors. Considering the radial and axial clearance fit between the bearing outer ring and the pedestal, the six-degree-of-freedom (6-DOF) frictional contact relationship between the outer ring and the pedestal is described, and a novel clearance fit model is proposed. In addition, the quasi-static models of deep groove ball bearings (DGBBs), angular contact ball bearings (ACBBs), and cylindrical roller bearings (CRBs) are established with the effects of rotational speed. The clearance fit model, quasi-static models of the bearings, and finite element (FE) model of the spindle are coupled to develop a dynamic model of a spindle-bearing system with pedestal tilt error. The contact performances of the DGBB and the vibration behaviors of the system with pedestal tilt errors are analyzed by simulation and experiment. The results show that the amplitude of the time-domain acceleration response decreases first and then increases with the increase of the pedestal tilt degree. The pedestal tilt error increases the contact angle of the DGBB, so the varying compliance (VC) vibration frequency value and its error also increase.

Investigating the effect of shot peening parameters on the peened surface damage using FEA

Shot peening is a cold working process that generates compressive residual stresses on the surface by blasting a numerous of shots at a very high speed, also it is considered a complex process to be simulated. In this study, a simulation of a single spherical shot on stainless steel 410 martensitic target was implemented utilizing a finite element software (ANSYS Explicit), to study the contact parameters, surface damage, and the generated residual compressive stresses (RCS) just beneath the surface. Several conditions were conducted varying the speed of the shot and the angle of incidence. Two prior made empirical models based on vertical shooting were used to calculate the depth and the diameter of indentation, then the calculated values were compared with the simulation's results. The novelty of this work is to test the validity of these quasi-static empirical models which are based on vertical actual shot peening and indentation experiments and use them to support inclined shot peening simulation as substitution for experiments. The results show that the validity for the empirical models which are utilized in prediction of inclined shot peening contact parameters was proved, inclined impact gives a better RCS beneath the surface, and less geometrical deformation which keep the surface roughness in an acceptable range, however, shooting at angles lower than 45° can cause material removal, a rough surface, and less RCS.

The influence of asymmetric faces on low-velocity impact failure of CFRP/aluminum foam composite sandwich beams

The influence of asymmetric faces on dynamic failure of fully clamped CFRP/aluminum foam composite sandwich beams subjected to low-velocity impact was studied. Three kinds of geometrically asymmetric sandwich cross-sections with an equal area density were designed. The low-velocity impact experiments of the composite sandwich beams were carried out to explore the dynamic response and the failure modes. Initial failure modes of top face fracture, indentation and core shear were found. The final failure modes, including top face fracture, core shear, bottom face fracture at one end and debonding between core and bottom face, were observed. It is shown that the composite sandwich beams experience the elastic bending to large deflection. The impact load is strengthened and increases rapidly at the large deflection stage due to effect of the fully clamped boundary. Furthermore, the quasi-static theoretical models were employed to predict the dynamic response of the composite sandwich beams. the theoretical models can capture the elastic bending of the composite sandwich beams reasonably, while the initial failure loads are slightly underestimated. The comparisons of peak load and midspan deflection suggest that the asymmetric composite sandwich beams with a thinner top face have better impact resistance than other designs of the faces.

Thermo-poroelastodynamic response of a borehole in a saturated porous medium subjected to a non-hydrostatic stress field

A thermo-poroelastodynamic (TPED) model incorporating the thermo-osmosis and thermal-filtration phenomena is used to analyze the transient response of a borehole subject to a non-hydrostatic stress field. The coupled TPED model is developed on the basis of equations of motion, fluid flow, heat transfer and constitutive equations so as to couple the stress wave, pressure wave and thermal wave in early times. Poroelastodynamics (PED) and thermo-elastodynamics (TED) models correspond two particular cases of the current model. The borehole problem is decomposed into the superposition of an axisymmetric mode and an asymmetric mode, corresponding to axisymmetric and deviatoric loadings, respectively. The semi-analytical solutions in the Laplace transform space are inverted numerically to obtain time-dependent displacement, pore pressure, temperature and stresses. Comparisons are performed among different sub-models to examine the influence of thermal effect on the response of pore pressure and stresses as well as the influence of pore water on the response of stress components. A direct comparison among the dynamic, quasi-static and static models shows that inertial effect is very important in the early times and it contributes to the early-time wave behavior.

Mathematical effects of linear visco-elasticity in quasi-static Biot models

We investigate and clarify the mathematical properties of linear poro-elastic systems in the presence of classical (linear, Kelvin-Voigt) visco-elasticity. In particular, we quantify the time-regularizing and dissipative effects of visco-elasticity in the context of the quasi-static Biot equations. The full, coupled pressure-displacement presentation of the system is utilized, as well as the framework of implicit, degenerate evolution equations, to demonstrate such effects and characterize linear poro-visco-elastic systems. We consider a simple presentation of the dynamics (with convenient boundary conditions, etc.) for clarity in exposition across several relevant parameter ranges. Clear well-posedness results are provided, with associated a priori estimates on the solutions. In addition, precise statements of admissible initial conditions in each scenario are given.