Elastic Equilibrium Research Articles

Three-dimensional elasticity solutions for doubly-curved composite shells by extended differential quadrature method

This paper presents the three-dimensional (3D) stress analysis of doubly-curved composite shells with general boundary conditions using the strong sampling surfaces (SaS) formulation. The SaS method is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as unknown functions that leads to a non-conventional shell formulation, in which strain–displacement and stress–strain relationships are represented in terms of SaS displacements. This is due to the use of Lagrange polynomials in approximating displacements, strains and stresses in the thickness direction. The outer surfaces are not included into a set of SaS that makes it possible to uniformly minimize the error used by Lagrange interpolation. The strong SaS formulation, based on direct integration of elasticity equilibrium equations in the thickness direction by the extended differential quadrature (EDQ) method, can be applied efficiently for high-precision calculations of doubly-curved composite shells with clamped and free edges. This is because in the SaS/EDQ formulation, displacements, strains and stresses of SaS are interpolated in a rectangular domain specified in a curvilinear coordinate system using a Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also used as basis functions. The proposed approach deals with equilibrium equations in terms of SaS stresses, avoiding the integration of second order differential equations in terms of SaS displacements, that greatly simplifies the implementation of the EDQ method.

Comparative analysis of ternary TiAlNb interatomic potentials: moment tensor vs. deep learning approaches

Intermetallic titanium aluminides, leveraging the ordered γ-TiAl phase, attract increasing attention in aerospace and automotive engineering due to their favorable mechanical properties at high temperatures. Of particular interest are γ-TiAl-based alloys with a Niobium (Nb) concentration of 5–10 at.%. It is a key question how to model such ternary alloys at the atomic scale with molecular dynamics (MD) simulations to better understand (and subsequently optimize) the alloys. Here, we present a comparative analysis of ternary TiAlNb interatomic potentials developed by the moment tensor potential (MTP) and deep potential molecular dynamics (DeePMD) methods specifically for the above mentioned critical Nb concentration range. We introduce a novel dataset (TiAlNb dataset) for potential training that establishes a benchmark for the assessment of TiAlNb potentials. The potentials were evaluated through rigorous error analysis and performance metrics, alongside calculations of material properties such as elastic constants, equilibrium volume, and lattice constant. Additionally, we explore finite temperature properties including specific heat and thermal expansion with both potentials. Mechanical behaviors, such as uniaxial tension and the calculation of generalized stacking fault energy, are analyzed to determine the impact of Nb alloying in TiAl-based alloys. Our results indicate that Nb alloying generally enhances the ductility of TiAl-based alloys at the expense of reduced strength, with the notable exception of simulations using DeePMD for the γ-TiAl phase, where this trend does not apply.

DSM design of cold-formed steel fixed-ended lipped channel columns undergoing distortional-global interaction

This work reports a numerical investigation dealing with the post-buckling behaviour, strength, and Direct Strength Method (DSM) design of cold-formed steel fixed-ended lipped channel columns experiencing coupling between distortional and global (flexural-torsional) buckling − distortional-global (D–G) interaction. Various cross-section dimensions and lengths are considered, to ensure that the columns selected undergo different levels and types (“true”, “secondary-bifurcation distortional” or “secondary-bifurcation global”) of D–G interaction. The results presented and discussed, determined by means of Abaqus shell finite element geometrically and materially non-linear analyses, consist of elastic and elastic-plastic post-buckling equilibrium paths, failure loads and collapse modes. The steel material behaviour is deemed elastic-perfectly plastic and several yield stresses are considered, thus making it possible to cover a wide column D–G slenderness range. Particular attention is devoted to identifying the most detrimental initial geometrical imperfection shapes, in the sense that they lead to the lowest column failure loads. The numerical failure load data gathered are subsequently used to assess the merits of the available DSM-based design approaches developed to handle cold-formed steel columns undergoing D–G interaction. Since these design approaches are shown to be either inefficient and/or improvable, the above failure load data are also used to propose modifications/improvements aimed at achieving an efficient failure load prediction in the specific context of fixed-ended lipped channel columns. The success of this endeavour provides encouragement to the authors in their search for a safe, accurate and reliable DSM-based design approach capable of handling cold-formed steel columns with arbitrary cross-section shapes and/or end support conditions that fail in D–G interactive modes.

A reduced order variational spectral method for efficient construction of eigenstrain‐based reduced order homogenization models

AbstractReduced order models (ROMs) are often coupled with concurrent multiscale simulations to mitigate the computational cost of nonlinear computational homogenization methods. Construction (or training) of ROMs typically requires evaluation of a series of linear or nonlinear equilibrium problems, which itself could be a computationally very expensive process. In the eigenstrain‐based reduced order homogenization method (EHM), a series of linear elastic microscale equilibrium problems are solved to compute the localization and interaction tensors that are in turn used in the evaluation of the reduced order multiscale system. These microscale equilibrium problems are typically solved using either the finite element method or semi‐analytical methods. In the present study, a reduced order variational spectral method is developed for efficient computation of the localization and interaction tensors. The proposed method leads to a small stiffness matrix that scales with the order of the reduced basis rather than the number of degrees of freedom in the finite element mesh. The reduced order variational spectral method maintains high accuracy in the computed response fields. A speedup higher than an order of magnitude can be achieved compared to the finite element method in polycrystalline microstructures. The accuracy and scalability of the method for large polycrystals and increasing phase property contrast are investigated.

Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

– This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method’s effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.

Experimental characterization and constitutive modeling of thermoplastic polyurethane under complex uniaxial loading

This paper presents the testing and constitutive modeling of a Thermoplastic Polyurethane (TPU) compound used in commercial applications. The tested specimens were extracted directly from a TPU sphere used in check valves through water-jet cutting. The tests included tensile and compression tests under complex uniaxial loading protocols to capture different nonlinear phenomena, such as stress softening, hysteresis, relaxation, creep, and rate dependence. The material is modeled assuming a nonlinear elastic equilibrium path that may exhibit stress softening (i.e., Mullins effect), and a hysteretic viscoplastic response that presents rate dependence at three different time scales. To achieve this constitutive behavior, a Parallel Rheological Framework model is used. The nonlinear elastic equilibrium path is modeled using the generalized Yeoh hyperelastic model. The stress softening of the equilibrium path is modeled using the Ogden-Roxburgh damage model on the deviatoric response. The hysteretic viscous response is further split into three viscoplastic chains to represent time dependence at three different time scales in a decoupled way. Each viscoplastic chain is modeled using the Bergstrom-Boyce model with its standard evolution law of the creep strain. The model parameters were found using a stochastic optimization scheme to simultaneously fit all the considered tests. The outstanding agreement between the model and the experimental data across a wide range of loading scenarios provides additional insight into the time-dependent behavior and deformation mechanism of TPUs. Moreover, it shows that the mechanical behavior of these materials can be represented by decoupling the nonlinear viscoplastic behavior in different time scales.

Stress state of the Mars’ and Venus’ interior

It is shown that most of the epicenters of marsquakes are located in the zones of extension and fairly large shear stresses associated with the deviation of Mars from hydrostatic equilibrium. Non-hydrostatic stresses in the interior of Venus are calculated for two types of models: an elastic model and a model with a lithosphere of varying thickness (150–500 km) overlying a weakened layer that has partially lost its elastic properties. Numerical modeling of the system of elastic equilibrium equations for a gravitating planet is carried out with a step of 1°×1° in latitude and longitude up to a depth of 480 km – the first phase transition zone in the mantle. The topography and the gravitational field of the planet are the boundary conditions of the problem. Overall, the level of nonhydrostatic stress on Venus is not very high. On the surface and in the crust, the highest shear stresses are observed in the region of the Maxwell Monte on Ishtar Terra. Beneath the Maxwell Monte, shear stresses in the crust reach 80 MPa and compressive stresses, 125–150 MPa, depending on the model. Tensile stresses around this region are about 20 MPa. The highest tensile stresses occur in the regions beneath structures such as Lavinia Planitia, Sedna Planitia, and Aino Planitia.

Elastic Equilibrium of a Hollow Cylinder of Finite Length Under Axisymmetric Force Loading

Elastic Equilibrium of a Hollow Cylinder of Finite Length Under Axisymmetric Force Loading

A random elastic traffic equilibrium problem via stochastic quasi-variational inequalities

The aim of the paper is to introduce a random elastic traffic equilibrium problem in a Hilbert space setting. The equilibrium condition is expressed by a random extension of the elastic Wardrop principle. Its characterization with a stochastic quasi-variational inequality is proved. Under suitable assumptions, the existence of a random equilibrium distribution is established. Furthermore, a numerical scheme to compute the random elastic traffic equilibrium distribution is presented. Finally a numerical example is discussed.

Interfacial stresses of beams hybrid strengthened by steel plate with outside taper and FRP pocket

This study proposes a new hybrid strengthening method, for which the reinforced concrete (RC) beams are strengthened in flexure by externally bonding a steel plate with variable thickness and an FRP pocket with end-anchorage. To investigate the influence of the stress distribution along the bondline on the strengthening effect, this paper develops an interfacial stress analysis model for RC beams strengthened with the proposed method under arbitrary loads. Based on the assumptions of linear elasticity, deformation coordination and static equilibrium conditions, the expressions of normal and shear stresses at the steel-concrete and FRP-steel interfaces of the strengthened RC beams under different loads and boundary conditions are derived. Finally, four-point bending tests on RC beams strengthened in flexure by using the proposed method were carried out. The steel plate with a fillet is beneficial to prevent the end debonding since it can significantly reduce the interfacial stresses at the end of the steel plate. Also, the results show good agreement between the calculated and tested results, which verified the applicability of the proposed model.

In-situ live jacking technology of corner transmission tower

Aiming at the current situation that the foundation of the corner transmission tower is corroded, sunken, or the height of the tower line is insufficient, innovative low-cost technology is proposed to upgrade the foundation in-situ based on no power outages, and the analysis of the whole process of the in-situ live jacking construction of the transmission tower is studied in this paper. Firstly, through elastic mechanical equilibrium relationship, two form-finding methods of numerical analysis are derived and the results of several form-finding methods are analyzed and compared by a project example, among which the result error of the finite element method and the catenary method is negligible, while the error of the parabolic method is about 2% compared with the first two methods, but it can meet the engineering requirements. Secondly, the relevant laws of internal force and deformation of the lifting process are summarized, and a comparative analysis of theoretical and measured results verified the reliability and safety of the technology. Through the calculation program of the self-compiled parabolic method, the laws of tower line changes during the lifting process of the transmission tower are summarized, providing technical support for engineering practice and reference for related research.

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

This paper introduces a 3D equilibrium-based finite element formulation for reinforced concrete. The stress unknowns in the proposed formulation strongly satisfy the equilibrium equations throughout the entire volume, with the exception of the mesh interfaces where the rebars intersect. The concrete stress tensor is interpolated using tetrahedral piece-wise linear elements with statically admissible discontinuities, while the rebars are considered as 1D elements embedded into the concrete, intersecting the triangular mesh. Thus, the global equilibrium is ensured by writing for each face of the mesh the traction continuity equation, while including the rebar stress contribution for the intersected triangular faces. The elastic equilibrium formulation is then written into an equivalent optimization problem, extended to the elastoplastic case by simply adding semi-definite matrix constraints on the concrete stress tensor, corresponding to a Rankine or a truncated Mohr–Coulomb criterion. As for the rebars, they are considered to obey a 1D perfectly elastoplastic behavior. The present formulation is also developed for limit analysis, to directly obtain an estimate of the lower bound of the collapse load. The resulting semi-definite programming optimization problems are solved using an in–house interior point algorithm.

Pure-displacement formulation and bulk modulus propagation for topology optimization with pressure loads

This work proposes a pure-displacement formulation for topology optimization with pressure loads and the use of a continuous boundary propagation technique. This technique allows controlling the non-solid (fluid or void) constitutive behavior by disseminating the desired bulk modulus to holes in the solid domain. The novelty of the proposed approach is to enable the distribution of solid, fluid, and void simultaneously with one design variable field. A density-based topology optimization approach is considered, and a gradient-based interior point algorithm is used. The linear elasticity equilibrium equations are discretized by the finite element method and solved with a sparse direct solver. The sensitivities are calculated by the discrete adjoint method with automatic differentiation. The optimization problems encompass compliance minimization (in 2D and 3D) and a compliant mechanism design to maximize the output displacement. Three examples evaluate the proposed formulation: the internally-pressurized lid, the piston head, and the inverter-compliant mechanism. The results show that different designs are obtained by considering different constitutive behavior for the non-solid phase.

RESEARCH OF OSCILLATIONS DURING END-MILLING AND THEIR INFLUENCE ON THE FORMATION OF THE MACHINED SURFACE

Purpose. Study of oscillations that occur during end-milling and their influence on the formation of the processed surface.
 Research methods. The research was carried out using the experimental method, in which oscillograms of part oscillations were recorded with the allocation of cutting time during milling. Basic fragments of oscillograms were studied using the analytical method, on which the parameters characterizing the milling process were measured, and their relationship with the processed surface was determined.
 Results. When milling with low spindle rotation frequencies, with down and up direction of feed, the part is affected by various types of oscillations, which are characteristic of the first and second speed zones of oscillations. During up and down end-milling, only forced oscillations operate in the first high-speed oscillation zone. In the second speed zone during up milling, the accompanying natural vibrations of the technological system are superimposed on the forced oscillations. It has been experimentally and analytically proven that the resulting deviation from the position of elastic equilibrium of the first wave of the accompanying oscillations affects the pitch and height of the waviness of the processed surface, which confirms the connection between the dynamics of end-milling and shaping.
 Scientific novelty. The impact of accompanying free oscillations of the technological system during cutting on the formation of the processed surface is evaluated using parameters characterizing the dynamics of end-milling.
 Practical value. The obtained results prove the influence of the cutting speed on the amplitude of the accompanying free oscillations of the technological system and provide an opportunity to choose cutting modes that ensure vibration stability of milling.

Effective Stiffness of Thin-Walled Beams with Local Imperfections.

Thin-walled beams are increasingly used in light engineering structures. They are economical, easy to manufacture and to install, and their load capacity-to-weight ratio is very favorable. However, their walls are prone to local buckling, which leads to a reduction of compressive, as well as flexural and torsional, stiffness. Such imperfections can be included in such components in various ways, e.g., by reducing the cross-sectional area. This article presents a method based on the numerical homogenization of a thin-walled beam model that includes geometric imperfections. The homogenization procedure uses a numerical 3D model of a selected piece of a thin-walled beam section, the so-called representative volume element (RVE). Although the model is based on the finite element method (FEM), no formal analysis is performed. The FE model is only used to build the full stiffness matrix of the model with geometric imperfections. The stiffness matrix is then condensed to the outer nodes of the RVE, and the effective stiffness of the cross-section is calculated by using the principle of the elastic equilibrium of the strain energy. It is clear from the conducted analyses that the introduced imperfections cause the decreases in the calculated stiffnesses in comparison to the model without imperfections.

Asymptotic Behavior of Constrained Local Minimizers in Finite Elasticity

We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads.

Whisker Sensing by Force and Moment Measurements at the Whisker Base.

We address the theoretical question which forces and moments measured at the base of a whisker (tactile sensor) allow for the prediction of the location in space of the point at which a whisker makes contact with an object. We deal with the general case of three-dimensional deformations as well as with the special case of planar configurations. All deformations are treated as quasi-static, and contact is assumed to be frictionless. We show that the minimum number of independent forces or moments required is three but that conserved quantities of the governing elastic equilibrium equations prevent certain triples from giving a unique solution in the case of contact at any point along the whisker except the tip. The existence of these conserved quantities depends on the material and geometrical properties of the whisker. For whiskers that are tapered and intrinsically curved, there is no obstruction to the prediction of the contact point. We show that the choice of coordinate system (Cartesian or cylindrical) affects the number of suitable triples. Tip and multiple point contact are also briefly discussed. Our results explain recent numerical observations in the literature and offer guidance for the design of robotic tactile sensory devices.

Static analysis and boundary effect of FG-CNTRC cylindrical shells with various boundary conditions using quasi-3D shear and normal deformations theory

In this paper, static analysis and the stress concentration phenomenon of functionally graded carbon nanotubes reinforced composite cylindrical shells with various boundary conditions are explored by using higher-order shear-normal deformation theory and an analytical approach. Five types of distributions of reinforced carbon nanotubes are considered, that is, uniform and four kinds of functionally gradient distribution of carbon nanotubes along with the thickness of the shell. Effective material properties of carbon nanotubes reinforced composite are estimated to the rule of mixture. The governing equations for FG-CNTRC cylindrical shells with various boundary conditions are solved analytically using simple trigonometric series and the Laplace transform. Moreover, the transverse shear and normal stress are reconstructed using 3D linear elasticity equilibrium equations. The present model is validated by comparing its numerical results with those of published works, including the 3D exact model. Using higher-order shear-normal deformation theory, this article has focused on investigating the stress concentration phenomenon, the distribution of stress components in the near-boundary region, and also assessing the effects of various parameters on the stress state of the FG-CNTRC cylindrical shells.

Nonlocal integral elasticity for third-order small-scale beams

Small-scale beams are basic structural components of miniaturized electro-mechanical systems whose design requires accurate modeling of size effects. In this research, the size-dependent behavior of nonlocal elastic beams is investigated by adopting the stress-driven elasticity theory. Kinematics of beams is modeled by the Reddy variational third-order beam theory accounting for the effective distribution of shear stresses on cross sections without needing the evaluation of shear correction factors. Stress-driven integral elasticity is thus extended to third-order small-scale beams providing an equivalent constitutive formulation with boundary conditions. The relevant nonlocal elastic equilibrium problem is formulated and an analytical strategy is proposed to obtain closed-form solutions. The present approach is elucidated by solving some structural problems of current interest in Nanotechnology.

Dynamically actuated soft heliconical architecture via frequency of electric fields

Dynamic electric field frequency actuated helical and spiral structures enable a plethora of attributes for advanced photonics and engineering in the contemporary era. Nevertheless, leveraging the frequency responsiveness of adaptive devices and systems within a broad dynamic range and maintaining restrained high-frequency induced heating remain challenging. Herein, we establish a frequency-actuated heliconical soft architecture that is quite distinct from that of common frequency-responsive soft materials. We achieve reversible modulation of the photonic bandgap in a wide spectral range by delicately coupling the frequency-dependent thermal effect, field-induced dielectric torque and elastic equilibrium. Furthermore, an information encoder prototype without the aid of complicated algorithm design is established to analogize an information encoding and decoding process with a more convenient and less costly way. A technique for taming and tailoring the distribution of the pitch length is exploited and embodied in a prototype of a spatially controlled soft photonic cavity and laser emission. This work demonstrates a distinct frequency responsiveness in a heliconical soft system, which may not merely inspire the interest in field-assisted bottom-up molecular engineering of soft matter but also facilitate the practicality of adaptive photonics.