Nonlinear Elastic Deformation Research Articles

Modeling and Compensation for Angular Transmission Error in Harmonic Drive Gearings (1st Report)

The research deals with a novel modeling and compensation approach for the transmission error in harmonic drive gearings. This paper, as the first report, presents a mathematical modeling framework for nonlinear components in the gearing, where nonlinear friction, nonlinear elastic characteristic, and angular transmission errors are handled to provide a precise numerical simulator. In the modeling of transmission errors, physical phenomena due to nonlinear elastic deformations in micro-displacement region are especially dealt with, as well as the synchronous component which has been already discussed in a variety of conventional studies. Based on the analyses of the phenomena, the nonlinear elastic component can be mathematically formulated by applying a modeling approach for the rolling friction with hysteresis attributes. Numerical simulations and experimental results using a prototype show the effectiveness of the proposed modeling.

ポアソン比にみられる4位柔軟節構造体の非線形弾性変形挙動

ポアソン比にみられる4位柔軟節構造体の非線形弾性変形挙動

A low-dimensional model for the red blood cell

The red blood cell (RBC) is an important determinant of the rheological properties of blood because of its predominant number density, special mechanical properties and dynamics. Here, we develop a new low-dimensional RBC model based on dissipative particle dynamics (DPD). The model is constructed as a closed-torus-like ring of 10 colloidal particles connected by wormlike chain springs combined with bending resistance. Each colloidal particle is represented by a single DPD particle with a repulsive core. The model is able to capture the essential mechanical properties of RBCs, and allows for economical exploration of the rheology of RBC suspensions. Specifically, we find that the linear and non-linear elastic deformations of healthy and malaria-infected cells match those obtained in optical tweezers experiments. Through simulations of some key features of blood flow in vessels, i.e., the cell-free layer (CFL), the Fahraeus effect and the Fahraeus-Lindqvist effect, we verify that the new model captures the essential shear flow properties of real blood, except for capillaries of sizes comparable to the cell diameter. Finally, we investigate the influence of a geometrical constriction in the flow on the enhancement of the downstream CFL. Our results are in agreement with recent experiments.

Constitutive Modeling of Rubber Behavior with an Application to Dynamics of Rolling Tires

AbstractMechanical response of technical rubbers is described by nonlinear elastic deformation, damage and hysteretic behavior under quasi‐static cyclic loading conditions, while dynamic stiffening and viscous effects are predominant for high frequency analysis. For the computation of the vibration of rolling tires, a material description incorporating all of these effects is needed. A constitutive model for filled rubbers under low frequency excitation is presented. A good agreement between the computational results and experimental data is also presented here. Furthermore, an idea to extend the constitutive model into a broad frequency domain is suggested through the uncoupled kinematic response in linear and nonlinear part. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DFT-based FEM analysis of nonlinear effects on indentation process in diamond crystal

We apply a new framework of a finite-element method (FEM) analysis with constitutive relations based on density functional theory (DFT), as an efficient method to characterize the nonlinear and anisotropic elastic deformation of single-crystal diamond. In our scheme, the stress–strain relations are obtained during FEM analysis on the fly based on the plane-wave-based DFT total-energy calculations and their numerical database is simultaneously constructed, which enables us to obtain high-precision stress without any empirical parameters even under finite strained conditions. To check its validity and accuracy, the shear deformation behavior of diamond crystal is analyzed under the strained condition. Then we examine the nonlinear effects on the indentation deformation of diamond single crystal, by comparing the results from the DFT-based constitutive relations with those from the linear elastic ones.

A micromechanics-based elastoplastic damage model for granular materials at low confining pressure

This paper is devoted to the formulation of a micromechanics-based constitutive model for granular materials under relatively low confining pressure. The constitutive formulation is performed within the general framework of homogenization for granular materials. However, new rigorous stress localization laws are proposed. Some local constitutive relations are established under the consideration of irreversible thermodynamics. Macroscopic plastic deformation is obtained by considering local plastic sliding in a limit number of families of contact planes. The plastic sliding at each contact plane is described by a non-associated plastic flow rule, taking into account pressure sensitivity and normal dilatancy. Nonlinear elastic deformation related to progressive compaction of contacts is also taken into account. Material softening is described by involving damage process related to degradation of microstructure fabric. The proposed model is applied to some typical granular materials (sands). The numerical predictions are compared with experimental data.

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

Challenges and Progress in High‐Throughput Screening of Polymer Mechanical Properties by Indentation

AbstractDepth‐sensing or instrumented indentation is an experimental characterization approach well‐suited for high‐throughput investigation of mechanical properties of polymeric materials. This is due to both the precision of force and displacement, and to the small material volumes required for quantitative analysis. Recently, considerable progress in the throughput (number of distinct material samples analyzed per unit time) of indentation experiments has been achieved, particularly for studies of elastic properties. Future challenges include improving the agreement between various macroscopic properties (elastic modulus, creep compliance, loss tangent, onset of nonlinear elasticity, energy dissipation, etc.) and their counterpart properties obtained by indentation. Sample preparation constitutes a major factor for both the accuracy of the results and the speed and efficiency of experimental throughput. It is important to appreciate how this processing step may influence the mechanical properties, in particular the onset of nonlinear elastic or plastic deformation, and how the processing may affect the agreement between the indentation results and their macroscopic analogues.

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity I: Incompressible Materials

Let $${A=\{{\bf x} \in \mathbb{R}^n : a 0}$$ , be the region occupied by an incompressible, nonlinearly hyperelastic material in its reference configuration. Deformations $${{\bf u} : {\it A} \rightarrow \mathbb{R}^n}$$ are therefore isochoric maps and so satisfy the incompressibility constraint $${\rm det}\nabla{\bf u} =1.$$ The displacement and the mixed displacement/zero-traction boundary-value problems are considered. Here a displacement boundary condition of the form $${\bf u} ({\bf x} ) =\sigma {\bf x}$$ is prescribed on one of the boundary components (where σ > 0 is a given constant) and the displacement on the remaining boundary component is either prescribed in the same form (in the case of the pure displacement boundary-value problem) or left unspecified (in the case of the mixed boundary-value problem). In this paper isoperimetric arguments are used to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of (possibly non-symmetric) isochoric deformations of the region A.

Simulation of CO2-geosequestration enhanced coal bed methane recovery with a deformation-flow coupled model

Coal bed methane (CBM) recovery and CO2 sequestration into coal seams coupled with enhanced CBM recovery have been recognized as an economically effective and environmentally friendly technology to improve the utilization of coal reserves. However, implementation of CBM and CO2 enhanced CBM (CO2-ECBM) production involves complex deformation-flow interactions in the coal. These aspects and their fundamental understanding remain as major concerns for CBM/ECBM modeling. Increasing interest in CBM and potentially in CO2-ECBM technology requires accurate predictive modeling to minimize investment risks. This paper proposed a deformation-flow coupled model to address aspects of model improvement. This model was developed based on nonlinear elastic deformation mechanics and gas percolation theory and implemented using an established computer program named F-RFPA2D - 2D Flow-coupled Rock Failure Process Analysis code. The numerical simulations of this model were carried out according to a CO2 capture and sequestration (CCS) integrated underground coal gasification (UCG) process designed for Zhongliangshan coal mine in southwest China. The individual operations comprising (1) conventional CBM recovery and CO2 sequestration into coal and (2) the integrated operation of CBM recovery with CO2 enhancement were numerically investigated, respectively. The results show that CO2 sequestration into the coal bed promotes rapid transport of CBM towards the gas producer wells with a longer production period and can enhance coal bed methane recovery by up to 80% under the conditions of using this this study.

Non-linear damage rheology and wave resonance in rocks

We address various deformational aspects of damaged materials with theoretical analyses and numerical simulations based on a non-linear continuum damage model. Quasi-static simulations of damage accumulation under cyclic load reproduce the laboratory-observed increase in the difference between tensile and compressive elastic moduli with ongoing deformation beyond the elastic regime. Modelling of wave propagation effects reproduces the observed relations between the resonance frequency and wave amplitude. In agreement with laboratory experiments, the simulated resonant curves are asymmetric, with gradual decrease of wave amplitudes for frequencies higher than the resonance value and strong reduction for lower frequencies. The predicted shift of the resonance frequency with increasing wave amplitude under constant material damage is only a few per cent, whereas the resonance frequency shift associated with increasing material damage may reach tens of per cent. The results show that the employed continuum damage rheology model provides a self-consistent treatment for multiple manifestations of non-linear elastic and brittle deformation of solids.

On spherical nanoindentations, kinking nonlinear elasticity of mica single crystals and their geological implications

In this paper, we show, using cyclic spherical nanoindentation experiments, that the deformation mechanisms in mica, including basal plane ruptures and delaminations, can be explained by invoking the presence of mobile dislocation walls, and incipient and regular kink bands. Our results clearly show that the energy dissipated or that was stored during the deformation of muscovite depends critically on its previous deformation history and/or the pre-existing defect concentration. Once nucleated, the dislocation-based incipient kink bands are believed to be responsible for the nonlinear elastic deformation and hysteretic loops obtained during cyclic loading. Moreover, a model is presented to estimate the number and distribution of dislocations and the energy consumed in their motion under the indenter. From the model, we also estimate the critical resolved shear stress for the motion of basal plane dislocations under the indenter. The implications of this work can be extended beyond mica to understand the nonlinear hysteretic deformation in other geological formations dominated by layered minerals.

Thermally activated model for tensile yielding of pristine single-walled carbon nanotubes with nonlinear elastic deformation

A thermally activated model for evaluating the tensile yield strain of pristine single-walled carbon nanotubes (SWCNTs) is established. Using a parabolic function to accurately describe the dependence of stress on strain, we derive a yield function relating the yield strain to temperature, strain rate and tube length. The activation energy and activation area are then determined from the MD results of the tensile yield strain as a function of temperature. We find that the activation energies for armchair SWCNTs range from 7.18 to 11.94 eV, depending on the radius of SWCNTs. Analyses of activation area and MD results reveal that the nucleation of a critical defect, which leads to the failure of SWCNTs, grows from a single bond size at 300 K to almost twice the size at 2100 K. On the basis of activation parameters, our model can be used to predict the yield strain of SWCNTs under experimental conditions.

Modeling and compensation for angular transmission error in harmonic drive gearings

AbstractThis paper presents a novel modeling and compensation approach for the angular transmission error in harmonic drive gearings. In the modeling, physical phenomena of the transmission error due to nonlinear elastic deformations in micro‐displacement region are especially dealt with, as well as the synchronous component which has been discussed in a variety of conventional studies. On the basis of the analyses of the phenomena, the nonlinear elastic component is mathematically modeled by applying a modeling framework for the rolling friction with hysteresis attributes. The proposed transmission error model has been adopted to the positioning system as a model‐based feedforward compensation manner. Experimental results using a prototype show the effectiveness of the proposed modeling and compensation. Copyright © 2009 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior

We report the fatigue endurance of a multifunctional β type titanium alloy exhibiting nonlinear elastic deformation behavior. Compared with other β type titanium alloys possessing linear elasticity, the alloy shows a similar ratio of fatigue strength to yield strength but a much lower ratio to tensile strength for smooth specimens with different fatigue stress ratios. The finding shows for the first time that titanium alloys exhibiting unstable elastic and plastic deformation behavior are capable of long-term load-bearing applications.

Density-Functional-Theory-Based Finite-Element Analysis of Diamond Single Crystal

We apply a finite-element analysis method based on first-principles density functional theory, to evaluate the nonlinear large elastic deformation of single-crystal diamond. The stress-strain relations are obtained during finite-element analysis on the fly based on the first-principles calculations and their numerical database is simultaneously constructed, which enables us to obtain high-precision stress without any empirical parameters even under finite strained conditions. The shear strength and mechanical behavior of diamond crystal are analyzed under various stress conditions, and then the uniaxial deformation of a diamond-crystal pillar model is examined through the present analysis method.

Calibration of machine deformation for quasistatic fracture toughness testing at high temperatures

Quasi-static J testing at elevated temperatures can be simplified to a great extent if extensometry for load line displacement measurement can be avoided. Towards this, a procedure has been developed for calibrating the non-linear elastic machine deformation as a function of load, using data for a blank specimen deforming in the elastic regime. The calibration procedure is demonstrated for ambient temperature 653 and 803 K. For a modified 9Cr–1Mo steel under the normalised and tempered condition without/with subsequent thermal aging, the viability of the calibration method for evaluating J is demonstrated for the two elevated temperatures.

Computational aspects of a pseudo-elastic constitutive model for muscle properties in a soft-bodied arthropod

In this paper we discuss the computational implementation of a new constitutive model that describes the muscle properties in a soft-bodied arthropod. Qualitatively, the muscle tissues behave similar to particle-reinforced rubber and are capable of large non-linear elastic deformations, show a hysteretic behavior, and display stress softening during the first few cycles of repeated loading. Such behavior can be described by the framework of pseudo-elastic transversely isotropic hyperelasticity. The computational model assumes compressible overall response, and is based upon a multiplicative split of the deformation gradient tensor into volumetric and isochoric parts. Details regarding the implementation of the computational model in the context of an implicit finite element solution procedure are presented. In particular, an explicit expression is provided for the material tangent stiffness tensor. Results obtained utilizing the new implementation are also presented.

Uniform strength for large deflections of cantilever beams under end point load

Analysis is conducted for slender beams with a varying cross-section under large non-linear elastic deformation. A thickness variation function is derived to achieve optimal - constant maximum bending stress distribution along the beam for inclined end load of arbitrary direction. Closed form solutions are derived for the large deflections that correspond to the various loading conditions. The analysis is repeated for a beam with optimally varying width (for arbitrary end force) and the width variation function is also determined.

A Cosserat point element (CPE) for nearly planar problems (including thickness changes) in nonlinear elasticity

This paper is concerned with nonlinear elastic deformations of a body that is a right-cylinder bounded by two major non-planar surfaces that are located symmetrically relative to the body’s planar middle surface. Deformations which maintain this symmetry are called nearly planar since they allow non-uniform thickness changes. Here, a four node quadrilateral Cosserat point element (CPE) is developed for these nearly planar problems. This planar CPE has three degrees of freedom at each node: two associated with the standard in-plane nodal displacements and one associated nodal thickness changes. The planar CPE can model problems where the thickness is specified (including plane strain) and where the normal component of the nodal traction vanishes (modified plane stress). In particular, thickness changes are modeled and the nonlinear three-dimensional constitutive equation is used without plane stress type approximations. Example problems are considered which compare the response of the planar CPE with other element formulations in the computer codes ABAQUS, ADINA, ANSYS and FEAP.