Finite Deformation Kinematics Research Articles

Influences of surface effects and large deformation on the resonant properties of ultrathin silicon nanocantilevers

The purpose of the present work is to quantify the influences of the discrete nature, the surface effects, and the large deformation on the bending resonant properties of long and ultrathin 〈100〉 silicon nanocantilevers. We accomplish this by using an analytical semi-continuum Keating model within the framework of nonlinear, finite deformation kinematics. The semi-continuum model shows that the elastic behaviors of the silicon nanocantilevers are size-dependent and surface-dependent, which agrees well with the molecular dynamics results. It also indicates that the dominant effect on the fundamental resonant frequency shift of the silicon nanocantilever is adsorption-induced surface stress, followed by the discrete nature and surface reconstruction, whereas surface relaxation has the least effect. In particular, it is found that a large deformation tends to increase the nonlinear fundamental frequency of the silicon nanocantilever, depending not only on its size but also on the surface effects. Finally, the resonant frequency shifts due to the adsorption-induced surface stress predicted by the current model are quantitatively compared with those obtained from the experimental measurement and the other existing approach. It is noticed that the length-to-thickness ratio is the key parameter that correlates the deviations in the resonant frequencies predicted from the current model and the empirical formula.

Subdivision elements for large deformation of liquid capsules enclosed by thin shells

This paper presents a numerical method for studying the large deformation of a liquid capsule enclosed by a thin shell in a simple shear flow. An implicit immersed boundary method is employed for calculating the hydrodynamics and fluid–structure interaction effects. A thin-shell model for computing the forces acting on the shell middle surface during the deformation is described within the framework of the Kirchhoff–Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. The interpolation of the reference and deformed surfaces of the shell is accomplished through the use of Loop's subdivision surfaces. The resulting limit surface is C 1-continuous which significantly improves the ability of the method in simulating capsules enclosed by hyperelastic thin shells with different physical properties. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical, oblate spheroidal and biconcave capsules in the shear flow. In addition, different types of motion such as tank-treading, tumbling and transition from tumbling to tank-treading have been studied over a range of shear rates and viscosity ratios.

Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

The deformation of a liquid capsule enclosed by a thin shell in a simple shear flow is studied numerically using an implicit immersed boundary method. We present a thin-shell model for computing the forces acting on the shell middle surface during the deformation within the framework of the Kirchhoff-Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. For hyperelastic materials, the plane-stress assumption is used to compute the hydrostatic pressure and the incompressibility condition yields the thickness strain component and the corresponding change in the thickness. The stresses developing over the cross-section of the shell are integrated over the thickness to yield the stress and moment resultants which are then used to compute the forces acting on the shell middle surface. The immersed boundary method is employed for calculating the hydrodynamics and fluid-structure interaction effects. The location of the thin shell is updated implicitly using the Newton-Krylov method. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical and oblate spheroidal capsules in the shear flow.

Plastic Intermediate Configuration and Related Spatial Differential Operators in Micromorphic Plasticity

The finite deformation kinematics of micromorphic plasticity is discussed in the framework of multiplicative decomposition of the macro- and microdeformation gradient tensor, suggesting the introduction of a so-called plastic intermediate configuration for the micromorphic continuum. The geometrical structure of the plastic intermediate configuration and the micromorphic curvature tensors are elucidated by invoking the differential operator of the relative covariant derivative with respect to the plastic intermediate configuration. Micromorphic curvature tensors arise in a natural way by considering scalar-valued differences. The latter measure the deformation process and are required to be form-invariant with respect to the chosen configuration.

Quantifying the size-dependent effect of the residual surface stress on the resonantfrequencies of silicon nanowires if finite deformation kinematics are considered

There are two major objectives to the present work. The first objective is to demonstratethat, in contrast to predictions from linear surface elastic theory, when nonlinear, finitedeformation kinematics are considered, the residual surface stress does impact the resonantfrequencies of silicon nanowires. The second objective of this work is to delineate,as a function of nanowire size, the relative contributions of both the residual(strain-independent) and the surface elastic (strain-dependent) parts of the surface stress tothe nanowire resonant frequencies. Both goals are accomplished by using the recentlydeveloped surface Cauchy–Born model, which accounts for nanoscale surface stressesthrough a nonlinear, finite deformation continuum mechanics model that leads to thesolution of a standard finite element eigenvalue problem for the nanowire resonantfrequencies. In addition to demonstrating that the residual surface stress does impact theresonant frequencies of silicon nanowires, we further show that there is a strong sizedependence to its effect; in particular, we find that consideration of the residual surfacestress alone leads to significant errors in predictions of the nanowire resonant frequency,with an increase in error with decreasing nanowire size. Correspondingly, thestrain-dependent part of the surface stress is found to have an increasingly importanteffect on the resonant frequencies of the nanowires with decreasing nanowire size.

Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress

We utilize the recently developed surface Cauchy–Born model, which extends the standard Cauchy–Born theory to account for surface stresses due to undercoordinated surface atoms, to study the coupled influence of boundary conditions and surface stresses on the resonant properties of 〈 1 0 0 〉 gold nanowires with { 1 0 0 } surfaces. There are two major purposes to the present work. First, we quantify, for the first time, variations in the nanowire resonant frequencies due to surface stresses as compared to the corresponding bulk material which does not observe surface effects within a finite deformation framework depending on whether fixed/free or fixed/fixed boundary conditions are utilized. We find that while the resonant frequencies of fixed/fixed nanowires are elevated as compared to the corresponding bulk material, the resonant frequencies of fixed/free nanowires are reduced as a result of compressive strain caused by the surface stresses. Furthermore, we find that for a diverse range of nanowire geometries, the variation in resonant frequencies for both boundary conditions due to surface stresses is a geometric effect that is characterized by the nanowire aspect ratio. The present results are found to agree well with existing experimental data for both types of boundary conditions. The second major goal of this work is to quantify, for the first time, how both the residual (strain-independent) and surface elastic (strain-dependent) parts of the surface stress impact the resonant frequencies of metal nanowires within the framework of nonlinear, finite deformation kinematics. We find that if finite deformation kinematics are considered, the strain-independent surface stress substantially alters the resonant frequencies of the nanowires; however, we also find that the strain-dependent surface stress has a significant effect, one that can be comparable to or even larger than the effect of the strain-independent surface stress depending on the boundary condition, in shifting the resonant frequencies of the nanowires as compared to the bulk material.

Choice of objective rate in single parameter hypoelastic deformation cycles

Hill [Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; 1950] demonstrated that “infinitesimal-displacement theory, used in classical elastoplasticity, may no longer be valid in elastic–plastic analysis because the convected terms in the rate of change of the stress acting on material particle may then not be negligible” [Lee EH. Some anomalies in the structure of elastic–plastic theory at finite strain. In: Carroll MM, Hayes M. editors. Nonlinear effects in fluids and solids, New York: Plenum Press; 1996. p. 227–49]. From this we may deduce that•the elastic deformation part may have considerable influence on the total deformation, even when it is relatively small, i.e. we are confronted with the vital requirement of properly computing elastic deformations.•Further, finite deformation kinematics should be applied, i.e. they should take account of possibly large rotations, e.g. through a formulation in an Eulerian frame.Xiao et al. [Xiao H, Bruhns OT, Meyers ATM. Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate. Int J Plast 1999;15:479–520] gave the mathematical proof, that Bernstein’s consistency criterion [Bernstein B. Relation between hypo-elasticity and elasticity. Trans Soc Rheol 1960;4:23–8; Bernstein B. Hypoelasticity and elasticity. Arch Ration Mech Anal 1960;6:90–104] is fulfilled in a hypoelastic law of grade zero if, and only if, the objective logarithmic stress rate [Xiao H, Bruhns OT, Meyers A. Hypo-elasticity model based upon the logarithmic stress rate. J Elasticity 1997;47:51–68] has been applied. This proof is of complicated mathematical nature. Here, we compare several objective Eulerian stress rates of corotational and non-corotational type for the hypoelastic law cited above in closed single parameter deformation cycles. It is found that the logarithmic stress rate returns the element to its stress-free original state after the closed cycle, thus confirming the findings in Xiao et al. (1999). We show that for some other objective rates the errors are accumulating to considerable amounts after several cycles, even when the deformation in investigation is relatively small. Interestingly, for Jaumann stress rate, the error may vanish for specified deformation measures.

A geometric framework for the kinematics of crystals with defects

Presented is a general theoretical framework capable of describing the finite deformation kinematics of several classes of defects prevalent in metallic crystals. Our treatment relies upon powerful tools from differential geometry, including linear connections and covariant differentiation, torsion, curvature and anholonomic spaces. A length scale dependent, three-term multiplicative decomposition of the deformation gradient is suggested, with terms representing recoverable elasticity, residual lattice deformation due to defect fields, and plastic deformation resulting from defect fluxes. Also proposed is an additional micromorphic variable representing additional degrees-of-freedom associated with rotational lattice defects (i.e. disclinations), point defects, and most generally, Somigliana dislocations. We illustrate how particular implementations of our general framework encompass notable theories from the literature and classify particular versions of the framework via geometric terminology.

A Finite-Deformation Constitutive Model of Bulk Metallic Glass Plasticity

A constitutive model of bulk metallic glass (BMG) plasticity is developed which accounts for finitedeformation kinematics, the kinetics of free volume, strain hardening, thermal softening, rate-dependency and non-Newtonian viscosity. The model has been validated against uniaxial compression test data; and against plate bending experiments. The model captures accurately salient aspects of the material behavior including: the viscosity of Vitreloy 1 as a function of temperature and strain rate; the temperature and strain-rate dependence of the equilibrium free-volume concentration; the uniaxial compression stress-strain curves as a function of strain rate and temperature; and the dependence of shear-band spacing on plate thickness. Calculations suggest that, under adiabatic conditions, strain softening and localization in BMGs is due both to an increase in free volume and to the rise in temperature within the band. The calculations also suggest that the shear band spacing in plate-bending specimens is controlled by the stress relaxation in the vicinity of the shear bands.

On the Effectiveness of Collars as Buckle Arrestors for Offshore Pipelines

Collars are devices similar to thick-walled rings which are welded to two adjacent strings of pipe during installation through the J-lay method. They are designed to support the suspended portion of the pipeline between the seafloor and the laugthing tower while the downstream segment of pipe is welded to the line. Because they locally increase the rigidity of the line, these devices also have the potential to arrest an eventual propagating buckle and thus, in such event, limit the damage to a small portion of the line. In this paper, the effectiveness of this type of local reinforcement as a buckle arrestor is studied through a nonlinear finite element model capable to reproduce the buckle propagation in pipes and its engagement to collars. The model is based on finite deformation kinematics and properly treats the contact that develops in the collapsed pipe behind the propagating buckle. Pipe and collar materials are modeled as J2 flow theory solids with isotropic hardening. Numerical procedure and results are summarized here and then used to propose a simplified method to evaluate the effectiveness of collars as buckle arrestors.

Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a Petrov–Galerkin multiscale formulation

The finite element model for strain localization analysis developed in a previous work is generalized to the finite deformation regime. Strain enhancements via jumps in the displacement field are captured and condensed on the material level, leading to a formulation that does not require static condensation to be performed on the element level. A general evolution condition is first formulated laying out conditions for the continued activation of a strong discontinuity. Then, a multiscale finite element model is formulated to describe the post-bifurcation behavior, highlighting the key roles played by the continuous and conforming deformation maps on the characterization of the finite deformation kinematics of a localized element traced by a strong discontinuity. The resulting finite element equation exhibits the features of a Petrov–Galerkin formulation in which the gradient of the weighting function is evaluated over the continuous part of the deformation map, whereas the gradient of the trial function is evaluated over the conforming part of the same map. In the limit of infinitesimal deformations the formulation reduces to the standard Galerkin approximation described in a previous work. Numerical examples are presented demonstrating absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions.

Fully C1‐conforming subdivision elements for finite deformation thin‐shell analysis

We have extended the subdivision shell elements of Cirak et al. [18] to the finite-deformation range. The assumed finite-deformation kinematics allows for finite membrane and thickness stretching, as well as for large deflections and bending strains. The interpolation of the undeformed and deformed surfaces of the shell is accomplished through the use of subdivision surfaces. The resulting ‘subdivision elements’ are strictly C1-conforming, contain three nodes and one single quadrature point per element, and carry displacements at the nodes only. The versatility and good performance of the subdivision elements is demonstrated with the aid of a number of test cases, including the stretching of a tension strip; the inflation of a spherical shell under internal pressure; the bending and inflation of a circular plate under the action of uniform pressure; and the inflation of square and circular airbags. In particular, the airbag solutions, while exhibiting intricate folding patterns, appear to converge in certain salient features of the solution, which attests to the robustness of the method.

Dynamic performance of integral buckle arrestors for offshore pipelines. Part II: Analysis

In the second part of this study we present models for simulating the quasi-static and dynamic propagation and arrest of buckles in pipelines. The models are developed within the framework of the nonlinear finite element ABAQUS and are used to simulate the quasi-static and dynamic arrest experiments in Part I. They are based on finite deformation kinematics and properly treat the contact that develops in the collapsed tube behind the propagating buckle. In the quasi-static model, the tube and arrestor materials are modeled as J 2 flow theory solids with isotropic hardening. In the dynamic model, the rate dependence of SS-304 is assumed to exhibit an overstress power–law dependence. The pressurizing medium is assumed to be vacuum. The dynamic model is shown to reproduce accurately the conditions of steady-state buckle propagation as well as the dynamic engagement of a buckle with an arrestor. For buckle velocities of interest, the buckle profile was found to have sharpened considerably compared to the quasi-static one. The numerical results showed the same dynamic enhancement of arrestor performance observed in the experiments. When the much sharper dynamic buckle profile engages an arrestor, it initially induces to it and to the downstream tube significant reverse ovality. This tends to obstruct and delay the flattening of the arrestor and tube which is the cause of the crossing of arrestors with quasi-static efficiencies of less than 0.7. The net result is that a higher pressure is required to cross the arrestors when the buckle is running. As in the experiments, the biggest dynamic enhancement was found to occur for arrestors of efficiencies in the range of 0.35–0.6. Based on the results of this study it is concluded that quasi-static design procedures for integral buckle arrestors proposed previously are conservative.

Computational aspects of incrementally objective algorithms for large deformation plasticity

A methodology for computationally efficient formulation of the tangent stiffness matrix consistent with incrementally objective algorithms for integrating finite deformation kinematics and with closest point projection algorithms for integrating material response is developed in the context of finite deformation plasticity. Numerical experiments illustrate an excellent performance of the proposed formulation in comparison with other algorithms. Copyright © 1999 John Wiley & Sons, Ltd.

On the performance of integral buckle arrestors for offshore pipelines

This paper presents the results of a study on the effectiveness of integral buckle arrestors for offshore pipelines. A series of full scale experiments were conducted where the pressure at which buckles propagating quasi-statically crossed arrestors of various lengths and thicknesses was established. The crossover pressures were used to establish the parametric dependence of the arresting efficiency (as defined in Kyriakides and Babcock, ASME Journal of Pressure Vessel Technology 102 (1980) and Proceedings of Offshore Technology Conference (1979)) of such devices. Buckles penetrated the arrestor in two modes: the flattening mode, where the arrestor and downstream pipe collapse in the same manner as the incoming buckle and the flipped mode, where the sense of collapse of the downstream pipe is orthogonal to the incoming buckle. The mode switch occurred in the neighborhood of efficiency of 0.7. The process of quasi-static engagement of an arrestor by a propagating buckle, the temporary arrest of the buckle and the eventual crossing of the arrestor were simulated through a finite element model. The model is based on finite deformation kinematics, incorporates J 2-type plasticity with isotropic hardening, and allows for contact of the walls of the collapsed section of pipe upstream of the arrestor. The model was verified by simulating each of the 15 physical experiments conducted using the actual geometric and material characteristics of the test specimens. The crossover pressures of the simulations were within 5% from the measured values and the mode of crossover was predicted correctly as well. The model was subsequently used to extend the experimental parametric study of arrestor efficiency. Some limiting values of the parameters were established from the results and several design recommendations are made.

Kinematics of finite elastoplastic deformation

A model in which plastic deformation occurs as a result of crystallographic slip is used to derive a formula for decomposing a deformation gradient into parts attributable to elastic and plastic deformation processes. The dislocation kinematics are those in long use, for example by Rice (1971, J. Mech. Phys. Solids 19, 433) and by Hill and Havner (1982, J. Mech. Phys. Solids 30, 5). The analysis presented here differs from that given by these authors in that it concerns total deformation (and its elastic and plastic parts) rather than incremental deformation. The decomposition obtained is inherent in the physics of the deformation process, arising naturally when the spatial discreteness of the active slip planes is taken into account. Expressions for the decomposition components are obtained by means of volumetric averaging of contributions of elastic deformation and slip to the total deformation. The decomposition captures the effects of isoclinic orientation central to Mandel's (1973, Int. J. Solids Struct. 9, 725) theory, but without introduction of the intermediate configuration that arises when the deformation is assumed to be expressible as though the plastic and elastic deformations occurred sequentially. Because no intermediate configuration arises, the established objectivity principle applies without modification and the concept of elastic embedding that is essential to proper selection of temporal rates is implemented quite naturally. Although the new decomposition is motivated by dislocation-mechanical considerations, the result is a continuum-mechanical expression that can be used in other contexts in which the sites of inelastic deformation are spatially separated in the material.

On the geometry of slip and spin in finite plastic deformation

Abstract A convenient description of the geometry of slip and the associated kinematics of finite deformation in an elastic-plastic crystal is presented. The physical meaning of the plastic spin is discussed in detail. The problem of uniaxial tension of a single crystal under multiple slip is analyzed, and a simple technique for the calculation of the rotation of the slip systems during plastic flow is presented. The paper closes with a relevant discussion of the kinematics of finite elastic-plastic deformations of a continuum.

Inelastic constitutive equations of polycrystalline metals subjected to finite deformation part I: Effect of grain rotation

In this article, a set of inelastic constitutive equations of polycrystalline metals is derived by combining a finite deformation kinematics of single crystal component, and a shear stress-shear strain relation of slip system based on a thermoactivated motion of dislocation. Interactions among grains are incorporated by “constant deformation gradient assumption.” The forms of these equations are rather simple internal variable theory types. By using these equations, some fundamental effects of grain rotations on inelastic behaviors of polycrystalline metals in a finite deformation range under complex loading and elevated temperature conditions are demonstrated. Some comments are given on a problem of plastic spin tensor.

Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples

This is an introductory survey of some selected topics in finite elasticity. Virtually no previous experience with the subject is assumed. The kinematics of finite deformation is characterized by the polar decomposition theorem. Euler’s laws of balance and the local field equations of continuum mechanics are described. The general constitutive equation of hyperelasticity theory is deduced from a mechanical energy principle; and the implications of frame invariance and of material symmetry are presented. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. Constitutive equations studied in experiments by Rivlin and Saunders (1951) for incompressible rubber materials and by Blatz and Ko (1962) for certain compressible elastomers are derived; and an equation characteristic of a class of biological tissues studied in primary experiments by Fung (1967) is discussed. Sample applications are presented for these materials. A balloon inflation experiment is described, and the physical nature of the inflation phenomenon is examined analytically in detail. Results for the different materials are compared. Two major problems of finite elasticity theory are discussed. Some results concerning Ericksen’s problem on controllable deformations possible in every isotropic hyperelastic material are outlined; and examples are presented in illustration of Truesdell’s problem concerning analytical restrictions imposed on constitutive equations. Universal relations valid for all compressible and incompressible, isotropic materials are discussed. Some examples of non-uniqueness, including that of a neo-Hookean cube subject to uniform loads over its faces, are described. Elastic stability criteria and their connection with uniqueness in the theory of small deformations superimposed on large deformations are introduced, and a few applications are mentioned. Some previously unpublished results are presented throughout.

Dynamic Plasticity

Recent advances in the understanding of the dynamic plastic response of crystalline solids are discussed. At the level of individual dislocations progress is being made on measurements of dislocation mobility at high stress levels and on elastodynamics solutions for transient dislocation motions. More progress is required on the understanding of changes in mobile dislocation density during dynamic plastic deformation. Widespread use of the Kolsky (or split-Hopkinson) bar has resulted in a reasonably clear picture of the dependence of flow stress on plastic strain rate for polycrystalline metals deformed at strain rates up to 103s−1. Influences of strain-rate history, temperature, and pressure require further investigation. At strain rates of approximately 103s−1’ to 105s−1 there is increasing evidence of a marked increase in flow stress with increasing strain rate. Pressure-shear plate impact experiments appear to be attractive for studying plastic response in this high strain-rate regime. Differences, if any, between stress-path effects at quasi-static strain rates and at strain rates of 103s−1 and higher remain poorly understood. Larger-than-predicted precursor decay observed in plate impact experiments on single crystals remains unresolved and of continuing fundamental interest; however, the importance of precursor decay measurements in determining dynamic plastic response appears to be diminishing because surface effects and limitations on the resolution of wave-front profiles represent serious constraints on the inferences that can be drawn from precursor decay measurements. Modeling of dynamic plastic response of polycrystals in terms of the response of slip systems is in an early stage of development. Kinematics of finite deformation by crystalline slip and consistent averaging techniques for modeling polycrystalline response are understood reasonably well. Increased emphasis on the understanding of the dynamic plastic response of single crystals and on the influence of microstructure appear desirable for sustained progress. Physically based models of wide applicability are required.