Elastostatic Analysis Research Articles

Elasto-Static Analysis of Composite Restorations in a Molar Tooth: A Meshless Approach.

Dental caries and dental restorations possess a long history and over the years, many materials and methods have been invented. In recent decades, modern techniques and materials have brought complexity to this issue, which has created the necessity to investigate more and more to achieve durability, consistency, proper mechanical properties, efficiency, beauty, good colour, and reduced costs and time. Combined with the recent advances in the medical field, mechanical engineering plays a significant role in this topic. This work aims at studying the elasto-static response of a human molar tooth as a case study, respecting the integral property of the tooth and different composite materials of the dental restoration. The structural integrity of the case study will be assessed through advanced numerical modelling resorting to meshless methods within the stress analysis on the molar tooth under different loading conditions. In this regard, bruxism is considered as being one of the most important cases that cause damage and fracture in a human tooth. The obtained meshless methods results are compared to the finite element method (FEM) solution. The advantages and disadvantages of the analysed materials are identified, which could be used by the producers of the studied materials to improve their quality. On the other hand, a computational framework, as the one presented here, would assist the clinical practice and treatment decision (in accordance with each patient's characteristics).

Elastostatic analysis of a module-based shape morphing snake-like robot

This paper describes the stiffness analysis of a module-based shape morphing snake-like robot. Snake-like robots have the characteristic of adapting to unstructured environments by exploiting their ability to reconfigure their body’s shape. However, the excellent mobility contrasts with the ability to transmit high loads, precluding its application in manufacturing operations. This article presents a hybrid structure based on reconfigurable modules equipped with lockable joints. The use of multiple modules in series allows for a large workspace. Furthermore, the parallel structure of the single modules provides for transferring or sustaining high loads. First, the reliability and precision of the theoretical model has been verified using finite element analysis (FEA). The relative errors are less than 5%. Then, a morphing module has been constructed as a physical demonstrator for the kinematic parameters and stiffness parameters used in elastostatic analysis. Finally, a five-segment prototype has been manufactured and tested. There is a deviation between the experimental results and the theoretical results due to manufacturing errors of the prototype but the trend of displacement change shown in the experimental results is basically consistent with the theoretical results.

On wave propagation in nanobeams

Wave propagation in Rayleigh nanobeams resting on nonlocal media is investigated in this paper. Small-scale structure-foundation problems are formulated according to a novel consistent nonlocal approach extending the special elastostatic analysis in Barretta et al. (2022). Nonlocal effects of the nanostructure are modelled according to a stress-driven integral law. External elasticity of the nano-foundation is instead described by a displacement-driven spatial convolution. The developed methodology leads to well-posed continuum problems, thus circumventing issues and applicative difficulties of the Eringen–Wieghardt nonlocal approach. Wave propagation in Rayleigh nanobeams interacting with nano-foundations is then analysed and dispersive features are analytically detected exploiting the novel consistent strategy. Closed form expressions of size-dependent dispersion relations are established and connection with outcomes available in literature is contributed. A general and well-posed methodology is thus provided to address wave propagation nanomechanical problems. Parametric studies are finally accomplished and discussed to show effects of length scale parameters on wave dispersion characteristics of small-scale systems of current interest in Nano-Engineering.

Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods

For both linear and nonlinear analysis, finite element method (FEM) software packages, whether commercial or in-house, have contributed significantly to ease the analysis of simple and complex structures with various working conditions. However, the literature offers other discretization techniques equally accurate, which show a higher meshing flexibility, such as meshless methods. Thus, in this work, the radial point interpolation meshless method (RPIM) is used to obtain the required variable fields for a nonlinear elastostatic analysis. This work focuses its attention on the nonlinear analysis of two benchmark plate-bending problems. The plate is analysed as a 3D solid and, in order to obtain the nonlinear solution, modified versions of the Newton–Raphson method are revisited and applied. The material elastoplastic behaviour is predicted assuming the von Mises yield surface and isotropic hardening. The nonlinear algorithm is discussed in detail. The analysis of the two benchmark plate examples allows us to understand that the RPIM version explored is accurate and allows to achieve smooth variable fields, being a solid alternative to FEM.

Curvature-based low-dimensional parametric model for the elastostatics analysis of planar compliant mechanisms under large deflection

<p indent=0mm>Based on<bold> </bold>the approximation of the bending curvature by Bernstein polynomials, a novel parametric model suitable for large deflection analysis is proposed for studying planar compliant mechanisms with uniform slender beam members. The numerical formulation is also presented based on Gaussian quadrature and Newton-Raphson iterations. By considering the bending curvature and coefficients of the Bernstein polynomial as the deformation field and unknown parameters, respectively, the equilibrium equations of the compliant mechanisms with geometric nonlinearity were established based on the principle of the minimum total potential energy. Further, the proposed method effectively avoided the numerical error introduced by using the deformation displacement to derive and calculate strain energy and greatly reduced the smoothness requirement of the trial function. Hence, the novel model possesses the benefits of high modeling accuracy, fast convergence speed in the numerical calculation, and versatility for application to straight and initially curved beams. Meanwhile, the parameterized curvature solutions have great geometric and mechanical significance that allows not only to realize the unified kinematic modeling of rigid and flexible bodies but also to efficiently present the complex and rich deformation information of slender beams under large deflections. The improved performance of the developed models was demonstrated through some classic numerical examples.

Accuracy analysis of spatial overconstrained extendible support structures considering geometric errors, joint clearances and link flexibility

The extendible support structure (ESS) is designed for ensuring the antenna panel of the synthetic aperture radar in a precise configuration. Therefore, the accuracy of the ESS is critical to the overall success of the radar mission. However, its accuracy is significantly affected by geometric errors, joint clearances and link flexibility. Taking into account all deviation sources simultaneously, a novel deterministic method is proposed to predict the pose error of the overconstrained ESS. Firstly, based on the comprehensive kinematic model of revolute joints with radial and axial clearances, the algorithm of contact detection between the journal and the bearing is developed with combination of the contact conditions and structural constraints. Next, the condensed stiffness matrices that merge flexible links and adjacent joints together are developed for the clearance-affected kinematic pairs, and the matrix structural analysis is adopted to generate the global stiffness matrix of the ESS. Finally, the accuracy of the ESS is determined by the elastostatic analysis and the virtual work principle, and the proposed method is validated through comparison between the numerical predictions and the experimental results.

The natural neighbor radial point interpolation method in the Elasto-Static analysis of Honeycomb-Shaped foams

Cellular solid materials are progressively becoming more predominant in lightweight structural applications as more technologies realize these materials can be improved in terms of performance, quality control, repeatability and production costs, when allied with fast developing manufacturing technologies such as Additive Manufacturing. In parallel, the rapid advances in computational power and the use of new numerical methods, such as Meshless Methods, in addition to the Finite Element Method (FEM) are highly beneficial and allow for more accurate studies of a wide range of topologies associated with the architecture of cellular solid materials. Since these materials are commonly used as the cores of sandwich panels, in this work, two different topologies were designed — conventional honeycombs and re-entrant honeycombs — for 7 different values of relative density, and tested on the linear-elastic domain, in both in-plane directions, using the Natural Neighbor Radial Point Interpolation Method (NNRPIM), a newly developed meshless method, and the Finite Element Method (FEM) for comparison purposes.

Load adaptation through bone remodeling: a mechanobiological model coupled with the finite element method.

This work proposes a novel tissue-scale mechanobiological model of bone remodeling to study bone's adaptation to distinct loading conditions. The devised algorithm describes the mechanosensitivity of bone and its impact on bone cells' functioning through distinct signaling factors. In this study, remodeling is mechanically ruled by variations of the strain energy density (SED) of bone, which is determined by performing a linear elastostatic analysis combined with the finite element method. Depending on the SED levels and on a set of biological signaling factors ([Formula: see text] parameters), osteoclasts and osteoblasts can be mechanically triggered. To reproduce this phenomenon, this work proposes a new set of [Formula: see text] parameters. The combined response of osteoclasts and osteoblasts will then affect bone's apparent density, which is correlated with other mechanical properties of bone, through a phenomenological law. Thus, this novel model proposes a constant interplay between the mechanical and biological components of the process. The spatiotemporal simulation used to validate this new approach is a benchmark example composed by two distinct phases: (1) pre-orientation and (2) load adaptation. On both of them, bone is able to adapt its morphology according to the loading condition, achieving the required trabecular distribution to withstand the applied loads. Moreover, the equilibrium morphology reflects the orientation of the load. These preliminary results support the new approach proposed in this study.

Two- and three-dimensional size-dependent couple stress response using a displacement-based variational method

Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required C1 continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems. Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems. Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response.

Influence of spatially varying material properties on the bimoment normal and shear stresses by warping torsion of FGM beams

In this contribution, which is an extension of author’s research [1,2], an influence of the in two and three directions varying material properties on non-uniform torsion of the Functionally Graded Material (FGM) thin-walled beams is originally investigated. Based on the semi-analytical solution of the fourth order differential equation for non-uniform torsion, the local finite element equations of the twisted FGM beam are presented, considering the non-uniform torsion with effect of warping and secondary deformations due to the angle of twist. The warping part of the first derivative of the twist angle caused by the bimoment is considered as an additional degree of freedom at the beam nodes. The Multi-Layers Method (MLM) [3], and the Reference Beam Method (RBM) [4] are extended for homogenization of the spatially varying material properties and relevant stiffnesses in the real beam onto effective longitudinally varying ones in the homogenized beam for the load case of non-uniform torsion.The focus of the numerical investigation, with consideration of the warping and Deformation Effect due to the Secondary Torsional Moment (STMDE), is on elastostatic analysis of straight FGM beams with rectangular hollow cross-sections. The influence of the variation of the material properties with respect to two or three directions on the angle of twist and the bimoment normal and torsional shear stresses is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid finite elements. This FGM warping non-uniform torsion beam finite element can be used not only in torsional analysis of general thin-walled beams but also in the design of FGM sensors and actuators in mechatronics.

The numerical analysis of symmetric cross-ply laminates using the natural neighbour radial point interpolation method and high-order shear deformation theories

Composite structures are commonly analysed using the Finite Element Method (FEM). However, new accurate and efficient discrete numerical techniques have appeared recently – the meshless methods. Thus, this work uses a meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) – to perform an elasto-static analysis of composite laminated plates. Meshless methods only require an unstructured nodal distribution to discretize the problem domain. In order to numerically integrate the integro-differential equation from the Galerkin weak formulation, a background integration mesh is constructed using the Voronoï diagram. Then, the nodal connectivity is enforced using the ‘influence-cell’ concept and the shape functions are obtained. In this work, laminated composite plates are analysed using distinct equivalent single layer theories, considering different transverse high-order shear deformation laws. Thus, several third-order, exponential and trigonometric transverse shear deformation theories are combined with the NNRPIM to analyse the structural response of composite laminated plates. In the end, composite laminated plates are numerically analysed and the meshless solutions are compared with the analytical solution available in the literature. Therefore, this works contributes with new solutions for classic composite symmetric cross-ply laminated plates and provides a comparative study on the accuracy of some high-order shear deformation theories (HSDTs).

Elastostatic Stiffness Analysis for the US/UPS Parallel Manipulators

The virtual joint method (VJM) cannot calculate the strain energy stored in each rod. In order to solve the problem, a modeling method of the elastostatic stiffness was investigated for the UP/UPS parallel manipulators (PMs), taking the example of the 6-SPS PM. The modeling method was based on screw theory, Castigliano’s theorem, and strain energy (where U, P, and S, respectively, denote universal, prismatic, and spherical joints). First, the actuator and constraint wrenches of the mechanism were obtained by screw theory. Second, compact limb stiffness matrices were obtained in terms of strain energy and Castigliano’s second theorem. Finally, analytic expressions for the overall stiffness matrix of the mechanism and the amplitudes of the actuator force were obtained by adopting the virtual work principle and the balance equation for the mobile platform. All relative errors between the results of the analytical model and the finite element model are below 2%, which validates the effectiveness of the elastostatic stiffness model. The virtual work index was adopted to evaluate the stiffness performance of the mechanism, and the results show that the stiffness is not only related to position and orientation but also closely related to the directions of external loads. It is also demonstrated that the method has general adaptability for the stiffness analysis for the US/UPS PMs, laying the foundation for further reasonable dynamic design and optimization of such manipulators.

Torsional warping elastostatic analysis of FGM beams with longitudinally varying material properties

In this paper, the influence of torsional warping of the cross-sections of thin-walled twisted Functionally Graded Material (FGM) beams on their elastostatic behavior is investigated. The material properties are assumed to vary longitudinally. The variation is described by a polynomial. Secondary deformations, resulting from the angle of twist, are considered. The transfer matrix method serves as the analysis tool. The transfer relations are derived and used for establishing finite element equations for twisted FGM beams in local coordinate systems. The warping part of the first derivative of the twist angle, caused by the bimoment, is considered as an additional degree of freedom at the nodes of the beam elements. The numerical investigation is performed with or without consideration of the Secondary Torsional Moment Deformation Effect (STMDE). It is focused on elastostatic analysis of straight cantilever FGM beams with doubly symmetric open as well as closed cross-sections. The influence of the longitudinal variation of the material properties and of the secondary torsion moment on the deformation and stress state is investigated. The results are compared with the ones obtained by a very fine mesh of standard solid and shell as well as warping beam finite elements.

On the role of material properties in ascending thoracic aortic aneurysms

One of the obstacles standing before the biomechanical analysis of an ascending thoracic aortic aneurysm (ATAA) is the difficulty in obtaining patient-specific material properties. This study aimed to evaluate differences on ATAA-related stress predictions resulting from the elastostatic analysis based on the optimization of arbitrary material properties versus the application of patient-specific material properties determined from ex-vivo biaxial testing. Specifically, the elastostatic analysis relies the on the fact that, if the aortic wall stress does not depend on material properties, the aorta has to be statistically determinate. Finite element analysis (FEA) was applied to a group of nine patients who underwent both angio-CT imaging to reconstruct ATAA anatomies and surgical repair of diseased aorta to collect tissue samples for experimental material testing. Tissue samples cut from excised ATAA rings were tested under equibiaxial loading conditions to obtain experimentally-derived material parameters by fitting stress-strain profiles. FEAs were carried out using both optimized and experimentally-derived material parameters to predict and compare the stress distribution using the mean absolute percentage error (MAPE). Although physiological strains were below yield point (range of 0.08–0.25), elastostatic analysis led to errors on the stress predictions that depended on the type of constitutive model (highest MAPE of 0.7545 for Yeoh model and 0.7683 for Fung model) and ATAA geometry (lowest MAPE of 0.0349 for patient P.7). Elastostatic analysis needs better understanding of its application for determining aneurysm mechanics, and patient-specific material parameters are essential for reliable accurate stress predictions in ATAAs.

A GE/BC imbedded local polynomial collocation method for two dimensional multivariable problems

In this study, a meshless numerical method for 2D problems with 2 variables coupled together in 2 partial differential equations (PDEs) is proposed. This method employs the local polynomial approximation with the weighted-least-squares (WLS) approach. A very distinguishing feature of this method from other collocation methods is the governing equations are satisfied at boundary nodes as well as the boundary conditions. By doing so the strong-form collocation method would be more stable. The 2D elasto-statics analysis is chosen for demonstrating the performance. Three test cases are implemented and the results are compared with exact or analytic solutions. Very good agreements are found.

Nonlinear static and stability analysis of composite beams by the variational asymptotic method

The Variational Asymptotic Method (VAM) that is a powerful method for solving thin-walled beam problems is used for nonlinear static and stability analysis of composite beams. The finite difference and the finite elements methods are used for calculating elastic deformation of beams with clamped-free boundary conditions. The case of simply supported beams is also solved using linear and nonlinear formulations and the results are compared with each other. Next, the shooting method is used for performing nonlinear elasto-static analysis of cantilever beams. Finally by applying perturbations on the equilibrium equations and using an eigenvalue analysis, elastic stability of clamped-free composite beams is analyzed. The obtained results are compared with the available data in the literature that have been obtained using deformation analysis.

Fast BEM multi-domain approach for the elastostatic analysis of short fibre composites

Composite materials are usually treated as homogeneous when carrying out structural design. However, failure in these materials often originated at their heterogeneous microstructure or constituent...

A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates

Probabilistic analysis is becoming more important in mechanical science and real-world engineering applications. In this work, a novel generalized stochastic edge-based smoothed finite element method is proposed for Reissner–Mindlin plate problems. The edge-based smoothing technique is applied in the standard FEM to soften the over-stiff behavior of Reissner–Mindlin plate system, aiming to improve the accuracy of predictions for deterministic response. Then, the generalized nth order stochastic perturbation technique is incorporated with the edge-based S-FEM to formulate a generalized probabilistic ES-FEM framework (GP_ES-FEM). Based upon a general order Taylor expansion with random variables of input, it is able to determine higher order probabilistic moments and characteristics of the response of Reissner–Mindlin plates. The significant feature of the proposed approach is that it not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also overcomes the inherent drawbacks of conventional second-order perturbation approach, which is satisfactory only for small coefficients of variation of the stochastic input field. Two numerical examples for static analysis of Reissner–Mindlin plates are presented and verified by Monte Carlo simulations to demonstrate the effectiveness of the present method.

A novel method to determine position and orientation errors in clearance-affected overconstrained mechanisms

Reaction forces and torques in overconstrained mechanisms are tied to the elasticity of the system. External wrenches applied to a mechanism generate joint reactions that can be determined only employing the stiffness analysis. This paper describes a novel method to find nodal displacements and rotations of overconstrained mechanisms due to clearance-affected joints. Unlike the previous formulations, the proposed method requires a single elastostatic analysis and employs the Virtual Work Principle to find the displacements at each node of the mechanism. Internal wrenches are first determined and then introduced in contact models of joints to obtain relative displacements/rotations of the kinematic pairs. The case of the journal-bearing is recalled introducing two contact scenarios, without limiting the use of other contact models. The method is applied to the 3-RRR spherical parallel manipulator under different clearance and pose parameters. Finally, feasible applications are supplied: multiple nodal loads, influence of preload and flexible joints, tradeoff between flexibility and clearance errors, introduction of manufacturing error in joints’ axes. The method has been validated using an equivalent Adams© model. The results revealed in good accordance making the method suitable to evaluate the influence of clearance in overconstrained mechanisms.

A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials

This manuscript presents an extension of the recently-developed high order complete scaled boundary shape functions to model elasto-static problems in functionally graded materials. Both isotropic and orthotropic functionally graded materials are modelled. The high order complete properties of the shape functions are realized through the introduction of bubble-like functions derived from the equilibrium condition of a polygon subjected to body loads. The bubble functions preserve the displacement compatibility between the elements in the mesh. The heterogeneity resulting from the material gradient introduces additional terms in the polygon stiffness matrix that are integrated analytically. Few numerical benchmarks were used to validate the developed formulation. The high order completeness property of the bubble functions result in superior accuracy and convergence rates for generic elasto-static and fracture problems involving functionally graded materials.