Large Elastic Deformations Research Articles

Mullins effect and cyclic stress softening of filled elastomers by internal sliding and friction thermodynamics model

Elastomers are characterized by their ability to undergo large elastic deformation. Nevertheless, their behavior exhibits stress softening, hysteresis and cyclic softening. The first phenomenon, known as Mullins effect, is commonly assumed to be either the result of an evolution in the hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch or the result of irreversible damage in the material or combination of both. Hysteresis and cyclic stress softening are often considered as the result of the effect of stress relaxation. Based on the physical structure of filled elastomers, the present study shows that the Mullins effect, hysteresis and cyclic softening can be modeled by dissipative friction phenomena due to internal sliding of the macromolecular chains and to sliding of the connecting chains on the reinforcing filler particles. This implies that the three effects are in fact related to one single deformation process. The proposed analysis allows to identify the state variables and to build a thermodynamic potential which accounts for the nonlinearity of the material behavior and for a time independent hysteresis. The constitutive model is 3D. Written in a rate form it applies to complex loadings: monotonic, cyclic, random fatigue, etc. Filled elastomers hysteresis loops and cyclic softening are represented with no need to introduce neither damage nor viscosity. The model was implemented in a Finite Element software to simulate a metal/elastomer lap joint. Good agreement with experiment was achieved.

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.

1035 第一原理密度汎関数法に基づくダイヤモンド結晶体の弾性大変形挙動の有限要素解析(OS10.電子・原子・マルチシミュレーションに基づく材料特性評価(8:マルチフィジックス・高分子),オーガナイズドセッション)

1035 第一原理密度汎関数法に基づくダイヤモンド結晶体の弾性大変形挙動の有限要素解析(OS10.電子・原子・マルチシミュレーションに基づく材料特性評価(8:マルチフィジックス・高分子),オーガナイズドセッション)

Mini-Workshop: The Mathematics of Electro-Active Smart Materials

The mini-workshop The Mathematics of Electro-Active Smart Materials , was attended by 15 participants (7 from Europe, 4 from USA and 1 from Israel). This workshop contained a nice blend of researchers with various backgrounds: mathematics, physics, engineering science and chemistry. The interdisciplinarity of the invited participants was essential because smart materials, such as piezoceramics, magnetostrictives and electroactive polymers, are characterized by strong couplings between their mechanical properties and applied electric, magnetic or thermal fields. Therefore skills in several areas of science are needed to understand the complicated and highly nonlinear behavior of such materials. Science and technology have produced amazing developments in the design of electronics and machinery using smart materials. Some everyday items are already incorporating smart materials (cars, the International Space Station, eyeglasses, coffee pots, speakers, for example) and the number of applications is growing steadily. For example, electro-rheological (ER) and magneto-rheological (MR) materials are fluids that can experience a dramatic change in their viscosity. They can change from a thick fluid (similar to motor oil) to nearly solid substances within the span of a millisecond when exposed to a magnetic or electric field; the effect can be completely reversed just as quickly when the field is removed. MR fluids are being developed for use in car shock absorbers, damping of machine vibration, prosthetic limbs, exercise equipment, and surface polishing of machine parts, in particular. ER fluids have been developed mainly for use in clutches and valves, as well as in engine mounts where they are designed to reduce noise and vibration in vehicles. More recently, magneto- and electro-sensitive elastomers have been developed and commercialized. Such materials are capable of large elastic deformations and a rapid and significant change in their material properties on the application of a magnetic or electric field. The coupling in the material response is achieved and optimized by distributing within an elastomeric matrix micron size ferrous particles. Cross-linking may occur, for example, in the presence of an applied field, whereby particles form chain-like structures aligned along the applied field direction. The material is then characterized by two families of preferred direction and may result in highly controllable smart materials. Mathematical and computational methods pervade research, development, testing, and evaluation problems encountered by researchers in the field of smart materials. Furthermore, increasing demands are being placed on research in the mathematical sciences because of their fundamental roles in the analysis and modelling issues that arise in such complex materials. In this Mini-Workshop we have discussed recent mathematical and computational questions related to the modelling of smart materials used in structures, actuators and sensors from the point of view of material properties and control of mechanical properties. The first talk by G. A. Maugin was on overview of existing theories capable of describing the strong coupling between mechanical properties and applied fields in electro- and magneto-active smart materials. This overview was extremely interesting also because it was punctuated with historical notes and remarks. On the other hand M. Zrinyi gave an exhaustive review of the phenomenology of such materials and of experimental data. This was extremely useful for facilitating a clear identification of further key experiments that should be conducted in order to provide more comprehensive data for improving the modelling process. The other talks have been devoted to

The biomechanics of arterial elastin

Uniaxial mechanical experiments have shown that a neo-Hookean/Gaussian model is suitable to describe the mechanics of arterial elastin networks [Gundiah, N., Ratcliffe, M.B., Pruitt, L.A., 2007. Determination of strain energy function for arterial elastin: Experiments using histology and mechanical tests. J. Biomech. 40, 586–594]. Based on the three-dimensional elastin architecture in arteries, we have proposed an orthotropic material symmetry for arterial elastin consisting of two orthogonally oriented and symmetrically placed families of mechanically equivalent fibers. In this study, we use these results to describe the strain energy function for arterial elastin, with dependence on a reduced subclass of invariants, as W = W ( I 1 , I 4 ) . We use previously published equations for this dependence [Humphrey, J.D., Strumpf, R.K., Yin, F.C.P., 1990a. Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Eng. 112, 333–339], in combination with a theoretical guided Rivlin–Saunders framework [Rivlin, R.S., Saunders, D.W., 1951. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phil. Trans. R. Soc. A 243, 251–288] and biaxial mechanical experiments, to obtain the form of this dependence. Using mechanical equivalence of elastin in the circumferential and longitudinal directions, we add a term in I 6 to W that is similar to the form in I 4 . We propose a semi-empirical model for arterial elastin given by W = c 0 ( I 1 − 3 ) + c 1 ( I 4 − 1 ) 2 + c 2 ( I 6 − 1 ) 2 , where c 0 , c 1 and c 3 are unknown coefficients. We used the Levenberg–Marquardt algorithm to fit theoretically calculated and experimentally determined stresses from equibiaxial experiments on autoclaved elastin tissues and obtain c 0 = 73.96 ± 22.51 kPa , c 1 = 1.18 ± 1.79 kPa and c 2 = 0.8 ± 1.26 kPa . Thus, the entropic contribution to the strain energy function, represented by c 0 , is a dominant feature of elastin mechanics. Because there are no significant differences in the coefficients corresponding to invariants I 4 and I 6 , we surmise that there is an equal distribution of fibers in the circumferential and axial directions.

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.

Viscoelastic Thin Polymer Films under Transient Residual Stresses: Two-Stage Dewetting on Soft Substrates

A nonmonotonic, two-stage dewetting behavior was observed for spin coated thin viscoelastic polymer films on soft elastic substrates. At times shorter than the relaxation time of the polymer (t<tau_{rep}), dewetting generated deep trenches in the soft rubbery substrate which, in turn, almost stopped dewetting. At later stages (t>>tau_{rep}), dewetting accelerated, accompanied by an unstable rim. However, holes nucleated at t<tau_{rep} showed only this second-stage behavior. Our observations are attributed to large elastic deformations in the substrate caused by transient residual stresses within the film.

Topological sensitivity analysis in large deformation problems

The aim of the present work is to apply the topological sensitivity analysis (TSA) to large-deformation elasticity based on the total Lagrangian formulation. The TSA results in a scalar function, denominated topological derivative, that gives for each point of the domain the sensitivity of a given cost function when a small hole is created. An approximated expression for the topological derivative is obtained by numerical asymptotic analysis. Numerical results of the presented approach are considered for elastic plane problems.

Non-linear finite element implementation of micropolar hypo-elastic materials

In this paper, updated Lagrangian finite element formulations for large elastic deformation of micropolar hypo-elastic materials are presented. Using representation theorems of tensor functions, the general form of the constitutive equations for the micropolar hypo-elastic materials model are presented. The finite element formulations are based on the general form of the micropolar hypo-elastic constitutive equations in conjunction with Jaumann rate and a new rate called gyration rate. Gyration rate describes the deformation of the material in view of an observer attached to the micro-structure. An incrementally objective stress and couple stress update procedure is developed and its applicability is investigated based on the Jaumann and gyration rates. Two planar examples, based on the simplified forms of the proposed constitutive equations with Jaumann and gyration rates are investigated. In the first example, a plate with a small circular hole is analyzed and it is shown that the results of non-linear analysis converge to the linear micropolar elasticity for the case of infinitesimal deformations. In another example, 2D deformation of a cantilever beam which is modeled as a slender rectangle is investigated. The results of Jaumann and gyration rates are very close to each other, but an increase in the micropolar material parameters, particularly an increase in the length-scale parameter results in the reduction of the deformation of the beam. Also, it is shown that for a specimen with very small dimensions, e.g. in a micro-beam, the effect of the length-scale parameter is more sensible than that of a specimen with an ordinary or large dimensions. This means that micropolar effects become important for deformations taking place at small scales.

Thermal Effect on the Postbuckling Behavior of an Elastic or Elastoplastic Truss

Temperature rise may lead to strength degradation and stiffness deterioration of structures under fire conditions. The purpose of this paper is to theoretically study the thermal effect on the postbuckling behavior of an elastic or elastoplastic two-member truss, based on large-deformation elasticity considerations. Two kinds of loadings are considered, i.e., trusses under constant temperature but increas- ing loads, and trusses under constant loads but rising temperature. For the case with constant temperature, the critical load of an elastic truss will be greatly reduced if the effect of yielding is taken into account. Moreover, yielding of material can cause the truss to bifurcate from the original elastic path. For the case with constant loads, a critical temperature that occurs as the limit point of the temperature- deflection curve can always be found. Besides, the presence of yielding can drastically reduce the critical temperature of an elastic truss, causing it to collapse in an abrupt manner. The solutions presented herein can be used as benchmarks for calibration of the accuracy of general finite-element procedures in analyzing structures under fire conditions.

F-bar projection method for finite deformation elasticity and plasticity using NURBS based isogeometric analysis

This paper presents projection methods to treat the incompressibility constraint in large deformation elasticity and plasticity within the framework of Isogeometric Analysis. The continuity property of higher-order Non-Uniform Rational B-Splines (NURBS) is explored in nearly incompressible applications and shown to produce accurate and robust results. A new nonlinear \(\mathop F\limits^ - \) projection method, based on a modified minimum potential energy principle and inspired by the \(\mathop B\limits^ - \) method is proposed for the large-deformation case. It leads to a symmetric formulation for which the consistent linearized operator for fully nonlinear elasticity is derived and used in a Newton-Raphson iterative procedure. The performance of the methods is assessed on several numerical examples, and results obtained are shown to compare favorably with other published techniques.

Formula omitted] and [formula omitted] projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements

This paper presents projection methods to treat the incompressibility constraint in small- and large-deformation elasticity and plasticity within the framework of Isogeometric Analysis. After reviewing some fundamentals of isogeometric analysis, we investigate the use of higher-order Non-Uniform Rational B-Splines (NURBS) within the B ¯ projection method. The higher-continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results. A new non-linear F ¯ projection method, based on a modified minimum potential energy principle and inspired by the B ¯ method is proposed for the large-deformation case. It leads to a symmetric formulation for which the consistent linearized operator for fully non-linear elasticity is derived and used in a Newton–Raphson iterative procedure. The performance of the methods is assessed on several numerical examples, and results obtained are shown to compare favorably with other published techniques.

On universal relations in orthotropic material subclasses

Universal relations that hold true for specific classes or subclasses of elastic materials form an integral part of the characterisation process for any material that can deform elastically. On the other hand, it has become understood that any class of hyper-elastic materials possessing anisotropic properties more complicated to those of transverse isotropy obeys no universal relations. However, there are met in practice special cases (subclasses) of anisotropic hyper-elastic materials that, under certain circumstances, may still obey a certain number of universal relations. Material characterisation purposes suggest therefore that there is a practical purpose in identifying and classifying different subclasses of anisotropic hyper-elastic materials that may obey any kind of universal relations and, hence, to derive and gradually build up catalogues or tables of different universal relations that such material subclasses obey when subjected to different deformation patterns. This paper heads towards this research direction by initially considering the orthotropic material class which is alternatively identified as the hyper-elastic crystal class of rhombic symmetry [A.E. Green, J.E. Adkins, Large Elastic Deformations, Clarendon Press, Oxford, 1960]. Then, for a certain set of appropriately chosen orthotropic and transverse isotropic hyper-elastic material subclasses, it develops and presents a considerable number of relevant universal relations, most of which are new in the literature.

Bending Deformation of C60 Nanowhiskers in Solution and Air

The mechanical properties of C60 nanowhiskers (C60NWs) grown in a solution by a liquid-liquid interfatical precipitation method were investigated by a bending method. C60NWs in the solution exhibited large elastic deformation where they recovered for large bending with a curvature of 290 μm. On the other hand, C60NWs in air were broken even for small bending with only a 650 μm curvature. From the analysis of the bending, the yield stress or breaking stress for C60NWs in air is estimated to be 7.7 MPa, that is larger by one order compared with 0.7 MPa for bulk C60 crystals. In addition, it is shown that the Young's modulus for C60NWs in the solution is 8.9 GPa, that is below a half of 20 GPa for bulk C60 crystals.

Large Deformability of 2D Framed Structures Connected by Flexible Joints

In this paper, we propose a finite displacement modeling technique based on the total Lagrangian formulation for the structural analysis of two-dimensional (2D) framed structures connected by flexible joints. Our proposed model consists of beam segments with three-node elements and flexible joints with multi-rotational nodes, which have higher rotational degrees of freedom for deformation compared with those of a classical continuum beam model. Therefore, the category of this model can be regarded as a piecewise C1 continuum beam model with rotational discontinuity on the joints. The objective of this study is to understand both the buckling modes and the post-buckling behaviors, which are key issues to understanding the integrity of the framed structures. Large elastic deformation simulations of the proposed model reveal that a 2D repetitive structure with finite square grids has biaxial symmetry buckling as a higher-order mode under uniaxial compression and has redundant post-buckling deformability, which rigid-jointed structures might not have.

Finite element model and experimental studies on rubber-cord composite

The rubber-cord composite (CRC) which is created of rubber matrix reinforced with textile cords is used for many applications such as pneumatic membranes, automobile tires, pneumatic air-springs, hydraulic hoses and many others. The CRC is characterized by strongly anisotropic material behaviour and can simultaneously undergo large elastic deformations. In this paper a finite element (FE) model was developed and applied to study the mechanical responses of CRC. This model consists of 8-node hexahedral brick elements describing rubber matrix and 3-D spar elements for modeling of textile cords. The experimental studies in uniaxial and cyclic tension were performed. The material constants of textile cords were fitted to experimentally measured data by approach technique using linear and bilinear elastic models. The simulations of uniaxial tensile tests using proposed FE model were carried out. The numerical results of simulations were compared to experimental ones in order to verify the accurateness of the FE model. The obtained results indicated that the proposed FE model can be applied for the modeling and simulation of mechanical behaviour of CRC.

On improved analytical method for stress–strain relationship for plate elements under axial compressive load

Accurate prediction of load carrying capacity of the ship hull is very important for its structural safety. The stress–strain relationship of hull structural element plays a vital role in this. A simple analytical method to predict the average stress–average strain relationship for plating between stiffeners in ship structures is proposed, in which two different methods are combined: the membrane stress method by Paik et al. (2000) involving large elastic deformation theory and the rigid plastic collapse mechanism theory by Yao et al. (1991). The former governs the stress–strain relationship up to ultimate strength while the latter is used beyond ultimate strength. A MATLAB code is developed for this purpose. The plating between stiffeners is analysed under axial load for different aspect ratio values. The present results are compared with published FEA results. It is concluded that the present method is quite accurate for deriving the average stress–average strain relationship of ship hull plating.

Assessing failure in microelectronic compounds

A strategy to estimate life expectation in copper vias is presented. To this end, two-scale finite-element simulations of thermal cycling are performed. The employed material model for the copper vias accounts for large elastic and plastic deformations and, additionally, for the growth of pores within the material. The common practice to extrapolate the accumulated plastic strain computed within a few steps of thermal cycling by means of a Coffin–Manson equation is critical examined. It is pointed out that a relatively large number of steps is necessary to obtain meaningful results. Furthermore, it is demonstrated that an extrapolation of the computed porosity up to a critical value allows for analogous conclusions.

A nonlinear elasticity model of macromolecular conformational change induced by electrostatic forces

In this paper we propose a nonlinear elasticity model of macromolecular conformational change (deformation) induced by electrostatic forces generated by an implicit solvation model. The Poisson–Boltzmann equation for the electrostatic potential is analyzed in a domain varying with the elastic deformation of molecules, and a new continuous model of the electrostatic forces is developed to ensure solvability of the nonlinear elasticity equations. We derive the estimates of electrostatic forces corresponding to four types of perturbations to an electrostatic potential field, and establish the existence of an equilibrium configuration using a fixed-point argument, under the assumption that the change in the ionic strength and charges due to the additional molecules causing the deformation are sufficiently small. The results are valid for elastic models with arbitrarily complex dielectric interfaces and cavities, and can be generalized to large elastic deformation caused by high ionic strength, large charges, and strong external fields by using continuation methods.