Large Elastic Deformations Research Articles

Isogeometric collocation for large deformation elasticity and frictional contact problems

Isogeometric collocation methods have been recently proposed as an alternative to standard Galerkin approaches as they provide a significant reduction in computational cost for higher-order discretizations . In this work, we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems. We first derive the non-linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization . Some numerical examples demonstrate the performance of collocation in its basic and enhanced versions, differing by the enforcement of Neumann boundary conditions . For problems with strong singularities , enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to Galerkin, whereas for weaker or no singularities enhanced and basic collocation may give very similar results. A large deformation contact formulation is subsequently developed and tested in the frictional setting, where collocation confirms the excellent performance already obtained for the frictionless case. Finally, it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three-dimensional setting with arbitrarily inclined non-matching discretizations, thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints.

An analysis of dynamics of risers during vessel motion by means of the rigid finite element method

The widespread use of lines, cables and risers in offshore applications requires research on the dynamic behavior of such systems. Accurate prediction of riser motion during operations is crucial. The models presented in the paper are used for static and dynamic analysis of planar slender systems undergoing large elastic deformations. Slender links are discretized by means of a modification of the rigid finite element method, which allows us to consider both bending and longitudinal flexibility. The method consists in dividing a link into rigid finite elements assuming inertial features connected by massless spring-damping elements; additionally, a spring element reflecting longitudinal flexibility is added. The formulation of the method presented allows us to take into account hydrodynamic forces and added mass. Correctness of the models and programs developed is proved by comparison of the results with an analytical solution and commercial software.When installing or moving a riser to a new site, any contact between it and another object should be prevented if possible. Thus, the numerical simulations presented in the paper concern the movement of a vessel with an attached riser (with and without fluid) in such a way that an obstacle on the seabed can be avoided.

Incremental Carcass Theory of Fibrous Media Under Large Elastic and Plastic Deformations

The two-level carcass theory as applied to fibrous media under large elastic and plastic deformations of matrix and fibers is expounded. The theory is incremental in its character and takes into account the deformation prehistory of the matrix and fibers in the composite media. At the macromechanical level of analysis, the geometrical equations of macroscopic deformations and the equations of macroscopic motion are defined in an incremental form. At the micromechanical (locally structural) level, incremental microboundary-value problems are solved for nodal material blocks on the basis of carcass deformations and their increments. From the internal fields found, the macroscopic stresses are determined, which allows one to close the system of equations of the macromechanical level of analysis.

Inverse finite element modeling of shells using the degenerate solid approach

The Inverse Finite Element Method (IFEM) for degenerate solid shells is introduced. IFEM allows determining the undeformed shape of a body (in this case, a shell-like body) such that it attains a desired shape after large elastic deformations. The model is based on the degenerate solid approach, which enables the use of the standard constitutive laws of Solid Mechanics. First, IFEM is applied to three popular benchmarks for validation purposes. Then, the capabilities of IFEM for inverse design are demonstrated by means of its application to the design of a microvalve.

Friction between silicon and diamond at the nanoscale

This work investigates the nanoscale friction between diamond-structure silicon (Si) and diamond via molecular dynamics simulation. The interaction between the interfaces is considered as strong covalent bonds. The effects of load, sliding velocity, temperature and lattice orientation are investigated. Results show that the friction can be divided into two stages: the static friction and the kinetic friction. During the static friction stage, the load, lattice orientation and temperature dramatically affects the friction by changing the elastic limit of Si. Large elastic deformation is induced in the Si block, which eventually leads to the formation of a thin layer of amorphous Si near the Si-diamond interface and thus the beginning of the kinetic friction stage. During the kinetic friction stage, only temperature and velocity have an effect on the friction. The investigation of the microstructural evolution of Si demonstrated that the kinetic friction can be categorized into two modes (stick-slip and smooth sliding) depending on the temperature of the fracture region.

An explicit, direct simulation of multiaxial finite strain inelastic behavior for polymeric solids

In contrast with the widely known elastoplastic behavior of hard solids with very small elastic deformations, we study elastoplastic behavior of soft solids dominated by very large elastic deformations. Toward this goal, thermo-coupled rate-independent and rate-dependent elastoplastic J2-flow models with evolving rubberlike elasticity are established for the first time in a sense of identically meeting the Clausius–Duhem inequality. Novel results are presented for coupling effects in three respects: (i) how finite strain elastic behavior may evolve with development of plastic flow; (ii) how plastic flow may be induced in a process of pure heating; and (iii) how strain rate effects may be characterized to ensure smooth transitions to the rate-independent case. It is demonstrated that complex inelastic deformation features observed in soft solids such as elastomers, including the Mullins effect, the permanent set, the induced anisotropy, the thermal recovery and the rate effect etc., may be derived as direct, natural consequences of the proposed model. In particular, explicit expressions for the constitutive functions incorporated are derivable from the uniaxial data for the purpose of achieving an explicit, exact simulation of the foregoing inelastic features, thus bypassing usual complexities both in choosing suitable forms of constitutive functions and in identifying unknown parameters in an approximate sense.

Temperature and strain-rate effects on the deformation behaviors of nano-crystalline graphene sheets

The deformation behavior of nanocrystalline graphene sheets is investigated by molecular dynamics (MD) simulation by coupling the effects of the temperature and strain rate. Mechanical deformation of graphene sheets, which is dominated by the competition between bond breaking and rotation, is essentially an atomic behavior. Similar to single-crystal graphene sheets, nanocrystalline graphene sheets usually exhibit bond breaking induced brittle fracture along grain boundaries after large elastic deformation. The elastic modulus decreases slightly with temperature as a result of softening but does not depend on the strain rate. A brittle-plastic transition by bond rotation and rearrangement under stress appears to occur at high temperature above 1000 K, but the ductility is unexpectedly reduced due to accelerated bond breaking. At small strain rates, it is easier for bonds to rearrange, vacancies to coalesce, and cracks to propagate in grain boundaries and plastic deformation with a larger activation volume occurs. However, at large strain rates, the relaxation time is too short for atomic bonds to rotate and rearrange under stress. Therefore, bond elongation and brittle fracture with a smaller activation volume takes place. The results demonstrate that the atomic behavior in grain boundaries is crucial to mechanical deformation in nanocrystalline graphene sheets, which is temperature and strain rate sensitive.

Filled elastomers: A theory of filler reinforcement based on hydrodynamic and interphasial effects

Experimental evidence has by now established that (i) the hydrodynamic effect and (ii) the presence of stiff interphases (commonly referred to as bound rubber) “bonding” the underlying elastomer to the fillers are the dominant microscopic mechanisms typically responsible for the enhanced macroscopic stiffness of filled elastomers. Yet, because of the technical difficulties of dealing with these fine-scale effects within the realm of finite deformations, the theoretical reproduction of the macroscopic mechanical response of filled elastomers has remained an open problem.The object of this paper is to put forward a microscopic field theory with the capability to describe, explain, and predict the macroscopic response of filled elastomers under arbitrarily large nonlinear elastic deformations directly in terms of: (i) the nonlinear elastic properties of the elastomeric matrix, (ii) the concentration of filler particles, and (iii) the thickness and stiffness of the surrounding interphases. Attention is restricted to the prominent case of isotropic incompressible elastomers filled with a random and isotropic distribution of comparatively rigid fillers. The central idea of the theory rests on the construction of a homogenization solution for the fundamental problem of a Gaussian elastomer filled with a dilute concentration of rigid spherical particles bonded through Gaussian interphases of constant thickness, and on the extension of this solution to non-Gaussian elastomers filled with finite concentrations of particles and interphases by means of a combination of iterative and variational techniques.For demonstration purposes, the theory is compared with full 3D finite-element simulations of the large-deformation response of Gaussian and non-Gaussian elastomers reinforced by isotropic distributions of rigid spherical particles bonded through interphases of various finite sizes and stiffnesses, as well as with experimental data available from the literature. Good agreement is found in all of these comparisons. The implications of this agreement are discussed.

New micro-structure designs of a polypropylene (PP) composite with improved impact property

In this paper, Hollow Glass Microspheres (HGMs) and Flexible Butadiene Nitrile Rubber Powders (BNRPs) were employed to jointly improve the impact property of polypropylene (PP). BNRPs and HGMs distributed in the PP matrix exhibited different modifying effects. Under the shocking force, large elastic deformation and shear yielding of flexible BNRPs occurred and the HGMs, which had spherical shape with smooth surface and were embedded in the PP matrix in advance, could arrest the crack propagation just as “Stop-Drilled Holes” (“SDHs”).

Large elastic deformation as a mechanism for soft seal leakage

Experiments were performed to visualize, in situ, the deformation and leakage of soft seals under differential pressure. Two annular seals and a flat, rectangular seal were investigated in the experiments. The observations show that large, nonuniform elastic deformation precedes seal leakage. A corresponding finite element model for a rectangular solid seal shows that large deformation leads to a decrease in the compressive sealing stress between the seal and the surface against which it seals. The predicted flow path for the model is qualitatively similar to the flow path observed in the experiments. A leakage mechanism in which nonuniform elastic deformation leads to seal leakage is proposed.

Nonlinear elastic behavior induced by nano-scale α phase in β matrix of β-type Ti–25Nb–3Zr–2Sn–3Mo titanium alloy

Large scale nonlinear elastic deformation behavior is induced by moderate precipitation of nano-scale α phase for Ti–25Nb–3Zr–2Sn–3Mo alloy after a strategy of cold rolling and short-time aging. Quantitive relationships are established between volume fraction of α phase (Vα) and elastic modulus, strength as well as maximal recoverable strain (εmax-R), where the highest εmax-R of 2.25% with strength of 900MPa corresponds to Vα of 10%. With the Vα increasing, the elastic behavior transforms from “double yielding” to nonlinear elasticity and then normal linearity. It is considered that the nonlinear elasticity of the studied (α+β) microstructure is probably related to “mechanical” stabilization of the alloy, i.e. a blocking effect of α phase on the lattice shear from body-centered cubic β phase to orthorhombic-α″ martensite.

Study on Leaf Spring Torsion with Elastic Large Deformation

The aim of this study is to show the relationship of elastic large deformation between torque and torsional angle for a bar with rectangular cross section. The formula about torque and torsional angle is modified by numerical analysis and experimental verification. High order item of torsional angle is added to the traditional formula, based on different boundary condition, which was set to zero before in the textbook. Using improved formula higher accuracy loading will be gotten when a leaf spring is twisted to elastic large deformation. Two types of leaf springs are used with depth of 6.8mm and 7.7mm with the same width of 100mm. The leaf springs are analyzed by finite element software of ANSYS with unit of SOLID 45. They are also tested by mechanical torsional loading device, with linear loading from 0 to 830N.m or 0 to 560N.m respectively. Both theoretical analysis and experimental results show that, there are some nonlinear char- acteristics when leaf springs with elastic large deformation. With comparison to linear fitting and cubic curve fitting, the linear fitting has larger error than cubic curve fitting. The cubic curve fitting can improve accuracy about 2 orders. In or- der to improve precision of leaf spring loading, the high-order curve should be used.

Nonlinear finite element analysis of functionally graded structures by enhanced assumed strain shell elements

In this work, large elastic deformation of structures made of functionally graded materials (FGMs) is investigated. Due to highly nonlinear nature of the problem, a total Lagrangian finite element formulation for a seven-parameter shell model is developed. The model accounts explicitly for thickness change, as well as normal stress and strain components through the thickness of the structure. The Green–Saint Venant strain tensor in conjunction with the Kirchhoff–Saint Venant constitutive equation is used to capture the geometric nonlinearity. To circumvent locking phenomenon at large distortions, an enhanced assumed strain formulation is adopted. Five different examples have been analyzed and the effect of enhanced strains and that of FGM exponent is investigated.

Dispersion characteristics of a nonlinear elastic metamaterial

We study wave dispersion in a one-dimensional nonlinear elastic metamaterial consisting of a thin rod with periodically attached local resonators. Our model is based on an exact finite-strain dispersion relation for a homogeneous solid, utilized in conjunction with the standard transfer matrix method for a periodic medium. The nonlinearity considered stems from large elastic deformation in the thin rod, whereas the metamaterial behavior is associated with the dynamics of the local resonators. We derive an approximate dispersion relation for this system and provide an analytical prediction of band-gap characteristics. The results demonstrate the effect of the nonlinearity on the characteristics of the band structure, including the size, location, and character of the band gaps. For example, large deformation alone may cause a pair of isolated Bragg-scattering and local-resonance band gaps to coalesce. We show that for a wave amplitude on the order of one-eighth of the unit cell size, the effect of the nonlinearity in the structure considered is no longer negligible when the unit-cell size is one-fourteenth of the wavelength or larger.

Optimal Control of Large Deformation Elasticity by Fiber Tension

AbstractObject of our interest is an elastic body Ω ⊂ ℝ3 which we can deform by applying a tension along certain given short fibers inside the body. The deformation of the body is desribed by a hyperelastic model with polyconvex energy density and a special energy functional for the tension along the fibers. We seek to apply (possibly large) deformations to the body so that a desired shape is obtained. To this end, we formulate an optimal control problem for the fiber tension field. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bioinspired legged-robot based on large deformation of flexible skeleton

In this article we present STARbot, a bioinspired legged robot capable of multiple locomotion modalities by using large deformation of its skeleton. We construct STARbot by using origami-style folding of flexible laminates. The long-term goal is to provide a robotic platform with maximum mobility on multiple surfaces. This paper particularly studies the quasistatic model of STARbot’s leg under different conditions. We describe the large elastic deformation of a leg under external force, payload, and friction by using a set of non-dimensional, nonlinear approximate equations. We developed a test mechanism that models the motion of a leg in STARbot. We augmented several foot shapes and then tested them on soft to rough grounds. Both simulation and experimental findings were in good agreement. We utilized the model to develop several scales of tri and quad STARbot. We demonstrated the capability of these robots to locomote by combining their leg deformations with their foot motions. The combination provided a design platform for an active suspension STARbot with controlled foot locomotion. This included the ability of STARbot to change size, run over obstacles, walk and slide. Furthermore, in this paper we discuss a cost effective manufacturing and production method for manufacturing STARbot.

FEM Simulation on Uniaxial Tension of Hyperelastic Elastomers

Modeling large nonlinear elastic deformation of elastomers is an important issue for developing new materials. Particularly, this is very promising for design and performance analysis of dielectric elastomers (DEs). These “smart materials” are capable of responding to an external electric field by displaying significant change in shape and size. In this paper, finite element method (FEM) was used to simulate the mechanical behavior of soft elastomers on uniaxial tension. Experimental data from uniaxial tensile tests were used in order to calibrate hyperelastic constitutive models of the material behavior. The constitutive model parameters were evaluated in ABAQUS/CAE. The 3D-model simulation results of a dumbbell shaped specimen at uniaxial tension shows very good correspondence with experimental data.

An updated Lagrangian method with error estimation and adaptive remeshing for very large deformation elasticity problems

SUMMARYAccurate simulations of large deformation hyperelastic materials by the FEM is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion. Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. However, it requires data transfer from the old mesh to the new one and this is a very delicate issue. In this paper, we present an updated Lagrangian formulation where the error is estimated and adaptive remeshing is performed in order to reach high level of deformations while controlling both the accuracy of the solution and mesh distortion. Special attention is given to data transfer methods and a very accurate cubic Lagrange projection method is introduced. A continuation method is used to automatically pilot the complete algorithm including load increase, error estimation, adaptive remeshing, and data transfer. A number of examples will be presented and analyzed. Copyright © 2014 John Wiley & Sons, Ltd.

Development of Hyperelastic Model for Natural Rubber Containing Weldlines

Several constitutive models of non-linear large elastic deformation based on strain-energy-density functions have been developed for hyperelastic materials. These models, coupled with the Finite Element Method (FEM), can effectively utilized by design engineers to analyze and design elastomeric products operating under the deformation states. However, due to the complexities of the mathematical formulation which can only obtained at the moderate strain and the assumption of material used for the analysis. Therefore it is formidable task for design engineer to make use of these constitutive relationships. In the present work, the strain-energy-density function of natural rubber part containing a weldline was constructed by using the Neural Network (NN) model. The analytical results were compared to those obtained by Neo-Hookean, Mooney-Rivlin, Ogden models. Good agreement between developed NN model and the existing experimental data was found, especially at very low strain and at very high strain.

전기화학적 에칭에 의한 AFM용 텅스텐 탐침의 강성 제어

− This paper presents a method of controlling the stiffness of a tungsten probe for an atomic force microscope (AFM) in order to provide high-quality phase contrast images in accordance with sample characteristics. While inducing sufficient deformation on sample surfaces with commercial Si or Si3N4 probes is difficult because of their low stiffness, a tungsten probe fabricated by electrochemical etching with appropriately high stiffness can generate relatively large elastic deformation without damaging sample surfaces. The fabrication of the tungsten probe involves two separate procedures. The first procedure involves immersing a tungsten wire with both ends bent parallel to the surface of an electrolyte and controlling the stiffness of the tungsten cantilever by decreasing its diameter using electrochemical etching in the direction of the central axis. The second procedure involves immersing the end of the etched tungsten cantilever in the direction perpendicular to the surface of the electrolyte and fabricating a tungsten tip with a tip radius of 20-50 nm via the necking phenomenon. The latter etching process applies pulse waves every 0.25 seconds to the manufactured tip to improve its yield. Finite element analysis (FEA) of the stiffness of the tungsten probe as a function of its diameter showed that the stiffness of the tungsten probes greatly varies from 56 N/m to 3501 N/m according to the cantilever diameters from 30 μm to 100 μm, respectively. Thus, the proposed etching method is effective for producing a tungsten probe having specific stiffness for optimal use with an AFM and certain samples.