Assumption Of Small Deformations Research Articles

IRROTATIONAL FLOW DUE TO FORCED OSCILLATIONS OF A BUBBLE

Abstract The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

Analysis of Amplification Effect and Optimal Control of the Toggle-Style Negative Stiffness Viscous Damper

This paper proposes a new toggle-style negative stiffness viscous damper (TNVD), and evaluates the performance of the TNVD with the displacement amplification factor (fd) and the energy dissipation factor (fE). Firstly, the composition and characteristics of the TNVD are introduced. Subsequently, the displacement amplification factor is introduced to evaluate the displacement amplification ability of the TNVD, and it is decomposed into a geometric amplification factor and an effective displacement coefficient. Then, based on the geometric amplification factor and effective displacement coefficient, the correlation between the TNVD’s displacement amplification ability and inter-story deformation is studied, and an improved TNVD is proposed. By the comparison of the finite element calculation results, it is found that the improved TNVD can utilize the assumption of small structural deformation. After that, the impacts of plentiful aspects, such as the length of the lower connecting rod, the horizontal inclination angle of the lower connecting rod, the inter-story deformation limit, the cross-sectional area of the connecting rod, the damping coefficient, and the negative stiffness on the fd and fE of the improved TNVD, are expounded. The research results show that when the length of the TNVD’s lower connecting rod remains unchanged, the fd and fE present a trend of increasing first and then decreasing with the increase in the horizontal inclination angle of the lower connecting rod. When the inter-story deformation is fixed, there exists an optimal lower connecting rod’s length that satisfies a specific relationship to achieve the optimal geometric amplification factor of the TNVD. By adjusting the damping parameters of the TNVD, we can obtain a better effective displacement coefficient greater than 0.95 in the proposed target region. Meanwhile, the fd and fE increase with the decrease in the negative stiffness. An optimization strategy for the improved TNVD has been proposed to ensure that the TNVD has the characteristics of operational safety, ideal displacement amplification capability, and energy dissipation capability. Furthermore, a multi-objective control design method with an additional improved TNVD structure is proposed. The vibration reduction effect of the structure with the improved TNVD and the effectiveness of the optimization strategy are verified through examples.

An automotive steel wheel digital twin for failure identification under accelerated fatigue tests

In this paper, a digital twin to predict failure in automotive steel wheels during experimental fatigue tests is proposed. The methodology involves a wheel finite element model that incorporates assembling and mounting processes, and CDTire/3D simulation package, employed to calculate tyre to rim generalised internal forces in several loading conditions. The two models are decoupled under small deformation assumption, which allows to reduce computational effort for batch simulations. The Digital Twin (DT) is designed to meet the standard requirements and the working principles of test benches for dynamic cornering, radial, and biaxial fatigue tests. The overall DT architecture is described with emphasis on different test bench logic and the introduction of simplifications to efficiently estimate failure under mean stress variation. Then, a simplified formulation of the critical plane McDiarmid multi-axial failure criterion is presented to face the non-proportionality of stress path under loading conditions. Finally, the methodology is validated against an industrial case study: the predicted failures are compared to experimental test result database for wheel topologies, material properties, test benches and loading conditions.

Meso-viscoelastic modeling of solid propellant using XCT technology and virtual element method

This paper proposes a scheme to develop 2D mesoscale propellant models from X-ray computed tomography (XCT) images, including digital image processing (DIP)-based models and molecular dynamics (MD)-based models. In the MD-based approach, parametric models can be constructed by presuming that the particles are discs that satisfy the size distribution and volume fraction derived from XCT images of propellant. Based on the advantages of the virtual element method (VEM) for simulating the deformation of composite materials, this study obtains statistical information from XCT images to generate mesostructural models and compare their mechanical behaviors. Under the assumption of small deformation, the mesoscopic behaviors predicted by both mesostructural models are in good agreement, indicating that simpler MD-based models are adequate to characterize the overall viscoelastic properties of propellant. The impacts of propellant mesostructure parameters were addressed by using MD-based models.

Structural topology optimization of three-dimensional multi-material composite structures with finite deformation

In this work, an explicit three-dimensional (3D) topology optimization approach is presented for multi-material composite structures accounting the finite deformation effect. The proposed method employs different sets of 3D Moving Morphable Voids (MMVs) to identify each phase material, resulting in explicit geometric descriptions of the optimized composite structures and a reduction in the number of design variables. The decoupling between the topology description and finite element analysis of composite structures enables the removal of redundant degrees of freedom, thereby mitigating the convergence issue of finite deformation analysis caused by low-density elements and leading to significant computational savings. Numerical experiments of composite structures with two- and three-phase materials are optimized to validate the accuracy and efficiency of the proposed method. The results demonstrate that, under finite deformation, the distribution of each phase material in optimized 3D composite structures is significantly affected by the amplitude of external loads, and the optimized layout could be quite different from its counterpart under small deformation assumption.

Correlation between parameters in the microstructural vector theory and Hill's plastic potential

The dissipation inequality provides a useful method for specifying constitutive equations of plastic deformation. There are two main ways to use the dissipation inequality to determine the plastic deformation path: by utilizing a plastic potential with the normality rule, or by not utilizing the plastic potential to determine the plastic flow according to the elastic distortion of the material coordinate axis (microstructural vectors). While both methods satisfy the inequality condition and predict experimental results well, they have been developed separately without discussion on relationship. This study presents a physics-based mathematical discussion on the relative dependence between the two methods of determining plastic flow and their point of connection, using Hill's quadratic plastic potential. For this, the conversion of the distortional length and angle changes in microstructural vectors to deviatoric strain tensor under small elastic deformation conditions is mathematically demonstrated. Using these equations, the rate of inelastic deformation derived from microstructural vectors exhibited a similar form to that derived from Hill's quadratic potential. Moreover, the model parameters of the microstructural vectors can be calibrated using the Hill's function. The plastic flow derived from the microstructural vectors and Hill's function exhibited a remarkably similar trend under the assumption of small elastic deformation, although not identical. The newly defined relationships were implemented numerically and validated through simulations. The parameters of the microstructural vector theory derived from Hill's plastic potential can be utilized in materials science and engineering.

Non-local effect of eccentrically simply supported beam on free vibration

Based on the Euler-Bernoulli beam theory, the coupling effect between bending vibration mode shape and longitudinal vibration mode shape of the beam is analyzed when the beam is supported by eccentric simply supported. Under the assumption of small deformation, the vibration control equations and the coupling boundary conditions are obtained through Hamilton's principle and the principle of virtual work variation. The numerical results under three different boundary conditions are given. It shows that the natural frequencies of the beam vary with the eccentricity distances are in complete agreement with the results obtained from finite element analysis and literature. The results strongly proved the validity and correctness of the coupling method in this paper. It also indicates that eccentric simply supported constraints have non-local effects.

A generalization of the Kelvin–Voigt model with pressure‐dependent moduli in which both stress and strain appear linearly

A generalization of the Kelvin–Voigt model that can represent viscoelastic materials whose moduli depend on the mechanical pressure is derived from an implicit constitutive relation in which both the Cauchy stress and the linearized strain appear linearly. For consistency with the assumption of small deformation, a thresholding approach is applied. The proposed mixed variational problem is investigated for its well‐posedness within the context of maximal monotone and coercive graphs. For isotropic extension or compression, a semi‐analytic solution of the generalization of the Kelvin–Voigt problem under stress control is presented. The corresponding numerical simulation for monotone and cyclic pressure loading is carried out, and the results then compared against the linearized model.

Active learning model for extracting elastic modulus of cell on substrate

The cell elastic modulus (Ec) is widely used as the mechanics-based marker to analyze the biological effects of substrates on cells. However, the employment of the Hertz model to extract the apparent Ec can cause errors due to the disobedience of the small deformation assumption and the infinite half-space assumption, as well as an inability to deduct the deformation of the substrate. So far, no model can effectively solve the errors caused by the above-mentioned aspects simultaneously. In response to this, herein, we propose an active learning model to extract Ec. The numerical calculation with finite element suggests the good prediction accuracy of the model. The indentation experiments on both hydrogel and cell indicate that the established model can efficiently reduce the error caused by the method of extracting Ec. The application of this model may facilitate our understanding about the role of Ec in correlating the stiffness of substrate and the biological behavior of cell.

A thermodynamically consistent viscoelastic with rate-dependent damage constitutive model

In this paper, a thermodynamically consistent viscoelastic and rate-dependent continuum damage constitutive model is proposed under small-deformation assumption. The model is consistent with quasi-static loading case that it is able to reproduce correct static strength when loading rate approaches to 0 by introducing a so-called pseudo damage state variable ϖ. In addition, the model can present increasing strength and strain at failure with increasing strain rate. The evolution of the pseudo damage state variable is described as a rate form of power of effective stress, while the damage state variable is given by a viscosity-like rate form equation. The viscoelastic behavior is described by the classic Prony series. Finally, the paper provides uniaxial tension, tension of plate-with-hole, and tension of particle-reinforced heterogeneous material as numerical examples to show viscoelastic behavior and damage development process.

Analysis of a landslide in sensitive clays using the material point method

Slope movements are generally classified into four different phases: pre-failure, failure, post-failure and eventual reactivation. In engineering applications, the pre-failure and failure phases are usually analysed using traditional numerical techniques, such as the finite-element method and the finite-difference method. However, these methods are often based on the assumption of small deformations and consequently are unsuitable for analysing the slope behaviour during the post-failure stage, which is usually characterised by very large deformations. To overcome this shortcoming, the material point method (MPM) is employed in the present study. Specifically, MPM is used to perform an analysis of a landslide in sensitive clays that occurred at Saint-Jude (Quebec, Canada) in 2010. To assess the accuracy of the analysis, the final profile and the displacement magnitude detected after the event are compared with those obtained by the numerical simulation. The results provided by MPM are in satisfactory agreement with field observations. The failure mechanism and the development of the failure surface within the slope are also reproduced successfully. These results also show that MPM is an attractive method for analysing the kinematics of landslides in sensitive clays, requiring also a limited number of conventional geotechnical parameters as input data.

Tube wall deformation behaviour of tensile-loaded blind-bolted connections in octagonal hollow section tubes

The structural behaviour of octagonal hollow section (OctHS) tubes under lateral tensile loads applied through blind-bolted T-stubs were experimentally and numerically investigated in this study. Two series of T-stub connections – single-side and double-side bolted T-stub connections, were tested to investigate the influence of the tube geometry and boundary conditions on the failure modes, deformation, and strength. Finite element (FE) models were developed and validated against the test results. Based on the validated FE models, parametric studies were conducted to further assess the influence of width-to-thickness ratios, tube length-to-width ratios and boundary conditions on the tube deformation and strength. Furthermore, the applicability of the tube deformation of 3% of a tube width to be the ultimate state criterion for OctHS tubes was evaluated using tube-face component shell FE models. A simplified two-dimensional analytical model was developed, and the calculation equations were proposed based on either small deformation assumption or large deformation assumption. The comparison of the strengths obtained from the proposed equations, numerical simulations and tests demonstrated that the proposed equations are able to give reasonable predictions on the yield and ultimate strengths of blind-bolted OctHS tube connections under tensile loading.

Dynamic modeling and analysis of thin-webbed spur gear pair

A dynamic model of spur gear pair combined with elastic shaft–disc–ring rotor is proposed to study the effect of web thickness on the dynamic characteristics. The shaft, disc and ring structures are simulated based on Timoshenko straight beam theory, Mindlin plate theory and Timoshenko curved beam theory, respectively. A universal method to derive the kinetic energy of rotating gear rotor is proposed based on multi-body dynamics theory and small deformation assumption. The elastic deformations of thin-web are included in the potential energy of time-varying mesh process. The equations of motion are established based on differential quadrature finite element method (DQFEM). The dynamic model of gear rotor is validated numerically by comparing to finite element (FE) model. The effects of web thickness on dynamic responses are studied by analyzing the dynamic deformations and stresses within web, which reveals that elasticity of web and shaft can reduce the vibration energy transferred to support structures, which contributes to vibration reduction of gear transmission system.

Seismic and failure behavior simulation for RC shear walls under cyclic loading based on vector form intrinsic finite element

Seismic performance and failure mode simulations for reinforced concrete (RC) structures are impeded by large deformations and material nonlinearity particularly when subjected to the combined action of compression, bending and shear. Structural deformation and failure analysis based on traditional finite element (FE) often fails for the small–deformation assumption and global equilibrium. Moreover, it remains difficult to obtain convergence results for the existing FE methods by applying general concrete constitutive models for the complicity of anisotropic materials. To simulate the hysteretic performance and the damage development process of the RC shear wall under cyclic loading, the vector form intrinsic finite element (VFIFE) method with independent particles was adopted and combined with the two-directional concrete constitutive model considering the loading and unloading rules to perform large deformation and damage estimations. Concrete planar triangular element and fiber element for steel bar equipped with the hysteretic constitutive models were used to model six RC shear walls with different design parameters. Owing to the merits of avoiding repeated accumulation of the global stiffness matrix for vector mechanics, the planar structural behaviors involving biaxial stress–strain and loading–unloading behavior were numerically demonstrated. The simulated results were in close agreement with the experimental tests, including the hysteretic loop, skeleton curve, bending–shear deformation, and damage mode. This research also provides successive evidence for damage process simulation by VFIFE for RC structures under seismic loading considering a comprehensive constitutive relation.

Dynamic Modeling of a Flexible-Link Flexible-Joint System with Tip Mass Considering Stiffening Effect

This paper presents the dynamic model of a flexible-link, flexible-joint manipulator system with a considerable stiffening effect of the flexible link. A gripper, along with tip mass, is attached at one end of the flexible link. By employing the extended Hamilton’s principle, a nonlinear governing equation of motion is derived along with several boundary constraints. Under the assumption of small deformation in free vibration, a simplified motion equation is deduced to determine the natural frequencies of the mechanical system. Four parameters of the system are selected to carry out the sensitivity study on frequency. The results show that the second frequency mainly depends on the mass of tip payload. Moreover, the third frequency is significantly affected by the moment inertia of tip payload. Regarding the constant angular motion, the finite element method is adopted to analyze the dynamic model by considering the stiffening effect. The frequency results are obtained which show a higher stiffness of the Single Flexible-link Flexible-joint (SFF) system with the angular velocity increasing. The influence of three factors (i.e., payload mass, length of the flexible link, and angular velocity) on the fundamental natural frequencies are discussed, which show instinct characteristics of the flexible manipulator system.

Thermal–Structural Coupling Analysis for Multiple Honeycomb Plates Connected by Hinges

The dynamical model of multiple honeycomb plates connected by hinges, including the thermal–structural coupling effect, is investigated in this paper. Under the assumption of small deformations and strains, the classical plate theory is used to establish the dynamical model of the honeycomb plate. The Chebyshev polynomial is adopted to describe the deformation of the plate and the linear torsional spring to represent the elastic hinge. Using the Rayleigh–Ritz method, the characteristic equation of the honeycomb sandwich structure is derived, from which the natural frequencies are obtained. Considering the thermal effect of the plate, the thermal–structural coupling dynamical equation is obtained. The thermal conduction in the thickness direction is calculated by the finite difference method. The present model is validated by comparing with the result generated by an equivalent orthotropic plate model established by the finite element (FE) software MSC Patran. The response analysis reveals the important effect of the metal skin and the hinge stiffness on the dynamic characteristics of the honeycomb sandwich structure. Numerical simulations reveal thermal–structural coupling features for the multi-honeycomb-plate structure from the perspective of the change in dynamic characteristics and the distribution of the temperature gradient. The thermal–structural coupling calculation method adopted herein can be used to solve the thermal conduction of the plate. It serves as the theoretical method for analyzing the dynamical behavior of the solar panel subjected to the solar heat.

Swinging and tumbling of multicomponent vesicles in flow

Biological membranes are host to proteins and molecules which may form domain-like structures resulting in spatially varying material properties. Vesicles with such heterogeneous membranes can exhibit intricate shapes at equilibrium and rich dynamics when placed into a flow. Under the assumption of small deformations and a two-dimensional system, we develop a reduced-order model to describe the fluid-structure interaction between a viscous background shear flow and an inextensible membrane with spatially varying bending stiffness and spontaneous curvature. Material property variations of a critical magnitude, relative to the flow rate and internal/external viscosity contrast, can set off a qualitative change in the vesicle dynamics. A membrane of nearly constant bending stiffness or spontaneous curvature undergoes a small amplitude swinging motion (which includes tangential tank-treading), while for large enough material variations the dynamics pass through a regime featuring tumbling and periodic phase-lagging of the membrane material, and ultimately for very large material variation to a rigid-body tumbling behaviour. Distinct differences are found for even and odd spatial modes of domain distribution. Full numerical simulations are used to probe the theoretical predictions, which appear valid even when studying substantially deformed membranes.

Double Scalar Variables Plastic-Damage Model for Concrete

A framework of the plastic-damage model with double scalar variables is established in nominal stress space under the small deformation assumption. In the damaged part, a damage tensor composed of double scalar variables is presented to comprehensively characterize the isotropic damage behavior in three-dimensional (3D) conditions. The damage laws of Young’s modulus and shear modulus are proposed to capture their different damage characteristic observed in the test. For one-dimensional (1D) and 3D conditions, the applicability of single scalar and double scalar damage variables is discussed. The macroscopic damage difference between these two damage variables when describing damage under 3D conditions is analyzed. In the plastic part, the plastic strain increment is determined by two parts of magnitude and direction. The magnitude is obtained by the consistency condition, and the flow direction is defined by the nonorthogonal flow rule that can satisfactorily reproduce the dilatancy behavior of concrete. The proposed model is implemented by the explicit Runge–Kutta (RK) method with the fifth-order accuracy and the Pegasus method. The performance of the model is assessed by the comparison results between the model and the cyclic loading and unloading test data under different stress paths.

Rayleigh-Wave Dispersion Analysis and Inversion Based on the Rotation.

Rotational observation is essential for a comprehensive description of the ground motion, and can provide additional wave-field information. With respect to the three typical layered models in shallow engineering geology, under the assumption of linear small deformation, we simulate the 2-dimensional radial, vertical, and rotational components of the wave fields and analyze the different characteristics of Rayleigh wave dispersion recorded for the rotational and translational components. Then, we compare the results of single-component inversion with the results of multi-component joint inversion. It is found that the rotational component has wider spectral bands and more higher modes than the translational components, especially at high frequencies; the rotational component has better anti-interference performance in the noisy data test, and it can improve the inversion accuracy of the shallow shear-wave velocity. The field examples also show the significant advantages of the joint utility of the translational and rotational components, especially when a low-velocity layer exists. Rotational observation shall be beneficial for shallow surface-wave exploration.

Simulation and Experiment Study on the Bridge Deflection Measuring Method Based on Secant Inclination

Deflection is the most direct indicator that reflects the bearing capacity of the bridge and the overall stiffness. There are many ways to measure the deflection of Bridges, and the inclination angle method is the most commonly used indirect method, but the existing theory of inclination angle method is relatively complicated. Based on the facts of the bridge small inclination, this article proposes the method of obtaining the bridge deflection by the inclination of the secant line constructed from the adjacent measurement points. Firstly, according to the bending deformation curve of general simply supported beam, the deflection calculation formula of each measuring point is derived based on the assumption of small deformation and the inclination Angle of measuring point. Secondly, a large commercial finite element software ANSYS 10.0 is used to carry out numerical simulation on the simply-supported beam under concentrated load in mid-span, and the deflection results of the numerical simulation are compared and verified with the theoretical results of the proposed method. Finally, the measured deflection results of the simply-supported beam model under mid-span load are compared with the theoretical results of the proposed method. The verification results show that if the actual model is consistent with the theoretical model, the proposed method has good accuracy.