Showing papers in "Journal of The Mechanics and Physics of Solids in 2004"

Mechanical properties of nanostructure of biological materials

Abstract: Natural biological materials such as bone, teeth and nacre are nanocomposites of protein and mineral with superior strength. It is quite a marvel that nature produces hard and tough materials out of protein as soft as human skin and mineral as brittle as classroom chalk. What are the secrets of nature? Can we learn from this to produce bio-inspired materials in the laboratory? These questions have motivated us to investigate the mechanics of protein–mineral nanocomposite structure. Large aspect ratios and a staggered alignment of mineral platelets are found to be the key factors contributing to the large stiffness of biomaterials. A tension–shear chain (TSC) model of biological nanostructure reveals that the strength of biomaterials hinges upon optimizing the tensile strength of the mineral crystals. As the size of the mineral crystals is reduced to nanoscale, they become insensitive to flaws with strength approaching the theoretical strength of atomic bonds. The optimized tensile strength of mineral crystals thus allows a large amount of fracture energy to be dissipated in protein via shear deformation and consequently enhances the fracture toughness of biocomposites. We derive viscoelastic properties of the protein–mineral nanostructure and show that the toughness of biocomposite can be further enhanced by the viscoelastic properties of protein.

Mechanics of deformation of single- and multi-wall carbon nanotubes

Abstract: An effective continuum/finite element (FE) approach for modeling the structure and the deformation of single- and multi-wall carbon nanotubes (CNTs) is presented. Individual tubes are modeled using shell elements, where a specific pairing of elastic properties and mechanical thickness of the tube wall is identified to enable successful modeling with shell theory. The incorporation and role of an initial internal distributed stress through the thickness of the wall, due to the cylindrical nature of the tube, are discussed. The effects of van der Waals forces, crucial in multi-wall nanotubes and in tube/tube or tube/substrate interactions, are simulated by the construction of special interaction elements. The success of this new CNT modeling approach is verified by first comparing simulations of deformation of single-wall nanotubes with molecular dynamics results available in the literature. Simulations of final deformed configurations, as well strain energy histories, are in excellent agreement with the atomistic models for various deformations. The approach was then applied to the bending of multi-wall carbon nanotubes (MWNTs), and the deformed configurations were compared to corresponding high-resolution images from experiments. The proposed approach successfully predicts the experimentally observed wavelengths and shapes of the wrinkles that develop in bent MWNTs, a complex phenomenon dominated by inter-layer interactions. Presented results demonstrate that the proposed FE technique could provide a valuable tool for studying the mechanical behavior of MWNTs as single entities, as well as their effectiveness as load-bearing entities in nanocomposite materials.

A unified treatment of strain gradient plasticity

Abstract: A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models a ...

A micro-macro approach to rubber-like materials—Part I: the non-affine micro-sphere model of rubber elasticity

Abstract: The contribution presents a new micro-mechanically based network model for the description of the elastic response of rubbery polymers at large strains and considers details of its numerical implementation. The approach models a rubber-like material based on a micro-structure that can be symbolized by a micro-sphere where the surface represents a continuous distribution of chain orientations in space. Core of the model is a new two-dimensional constitutive setting of the micro-mechanical response of a single polymer chain in a constrained environment defined by two micro-kinematic variables: the stretch of the chain and the contraction of the cross section of a micro-tube that contains the chain. The second key feature is a new non-affine micro-to-macro transition that defines the three-dimensional overall response of the polymer network based on a characteristic homogenization procedure of micro-variables defined on the micro-sphere of space orientations. It determines a stretch fluctuation field on the micro-sphere by a principle of minimum averaged free energy and links the two micro-kinematic variables in a non-affine format to the line-stretch and the area-stretch of the macro-continuum. Hence, the new model describes two superimposed contributions resulting from free chain motions and their topological constraints in an attractive dual geometric structure on both the micro- and the macro-level. Averaging operations on the micro-sphere are directly evaluated by an efficient numerical integration scheme. The overall model contains five effective material parameters obtained from the single chain statistics and properties of the network with clearly identifiable relationships to characteristic phenomena observed in stress–strain experiments. The approach advances features of the affine full network and the eight chain models by a substantial improvement of their modeling capacity. The excellent predictive performance is illustrated by comparative studies with previously developed network models and by fitting of various available experimental data of homogeneous and non-homogeneous tests.

Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension

Abstract: The membrane deflection experiment developed by Espinosa and co-workers was used to examine size effects on mechanical properties of free-standing polycrystalline FCC thin films. We present stress–strain curves obtained on films 0.2, 0.3, 0.5 and 1.0 μm thick including specimen widths of 2.5, 5.0, 10.0 and 20.0 μm for each thickness. Elastic modulus was consistently measured in the range of 53– 55 GPa for Au, 125– 129 GPa for Cu and 65– 70 GPa for Al. Several size effects were observed including yield stress variations with membrane width and film thickness in pure tension. The yield stress of the membranes was found to increase as membrane width and thickness decreased. It was also observed that thickness plays a major role in deformation behavior and fracture of polycrystalline FCC metals. A strengthening size scale of one over film thickness was identified. In the case of Au free-standing films, a major transition in the material inelastic response occurs when thickness is changed from 1 to 0.5 μm . In this transition, the yield stress more than doubled when film thickness was decreased, with the 0.5 μm thick specimen exhibiting a more brittle-like failure and the 1 μm thick specimen exhibiting a strain softening behavior. Similar plasticity size effects were observed in Cu and Al. Scanning electron microscopy performed on Au films revealed that the number of grains through the thickness essentially halved, from approximately 5 to 2, as thickness decreased. It is postulated that this feature affects the number of dislocations sources, active slip systems, and dislocation motion paths leading to the observed strengthening. This statistical effect is corroborated by the stress–strain data in the sense that data scatter increases with increase in thickness, i.e., plasticity activity. The size effects here reported are the first of their kind in the sense that the measurements were performed on free-standing polycrystalline FCC thin films subjected to macroscopic homogeneous axial deformation, i.e., in the absence of deformation gradients, in contrast to nanoindentation, beam deflection, and torsion, where deformation gradients occur. To the best of our understanding, continuum plasticity models in their current form cannot capture the observed size scale effects.

Stress-induced transformation behavior of a polycrystalline NiTi shape memory alloy: micro and macromechanical investigations via in situ optical microscopy

Abstract: An experimental investigation of the micro and macromechanical transformation behavior of polycrystalline NiTi shape memory alloys was undertaken. Special attention was paid to macroscopic banding, variant microstructure, effects of cyclic loading, strain rate and temperature effects. Use of an interference filter on the microscope enabled observation of grain boundaries and martensitic plate formation and growth without recourse to etching or other chemical surface preparation. Key results of the experiments on the NiTi include observation of localized plastic deformation after only a few cycles, excellent temperature and stress relaxation correlation, a refined definition of “full transformation” for polycrystalline materials, and strain rate dependent effects. Several of these findings have critical implications for understanding and modeling of shape memory alloy behavior.

Non-local crystal plasticity model with intrinsic SSD and GND effects

Abstract: A strain gradient-dependent crystal plasticity approach is presented to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. In order to be capable of predicting scale dependence, the heterogeneous deformation-induced evolution and distribution of geometrically necessary dislocations (GNDs) are incorporated into the phenomenological continuum theory of crystal plasticity. Consequently, the resulting boundary value problem accommodates, in addition to the ordinary stress equilibrium condition, a condition which sets the additional nodal degrees of freedom, the edge and screw GND densities, proportional (in a weak sense) to the gradients of crystalline slip. Next to this direct coupling between microstructural dislocation evolutions and macroscopic gradients of plastic slip, another characteristic of the presented crystal plasticity model is the incorporation of the GND-effect, which leads to an essentially different constitutive behaviour than the statistically stored dislocation (SSD) densities. The GNDs, by their geometrical nature of locally similar signs, are expected to influence the plastic flow through a non-local back-stress measure, counteracting the resolved shear stress on the slip systems in the undeformed situation and providing a kinematic hardening contribution. Furthermore, the interactions between both SSD and GND densities are subject to the formation of slip system obstacle densities and accompanying hardening, accountable for slip resistance. As an example problem and without loss of generality, the model is applied to predict the formation of boundary layers and the accompanying size effect of a constrained strip under simple shear deformation, for symmetric double-slip conditions.

Atomistic simulation of the structure and elastic properties of gold nanowires

Abstract: We performed atomistic simulations to study the effect of free surfaces on the structure and elastic properties of gold nanowires aligned in the 〈1 0 0〉 and 〈1 1 1〉 crystallographic directions. Computationally, we formed a nanowire by assembling gold atoms into a long wire with free sides by putting them in their bulk fcc lattice positions. We then performed a static relaxation on the assemblage. The tensile surface stresses on the sides of the wire cause the wire to contract along the length with respect to the original fcc lattice, and we characterize this deformation in terms of an equilibrium strain versus the cross-sectional area. While the surface stress causes wires of both orientations and all sizes to increasingly contract with decreasing cross-sectional area, when the cross-sectional area of a 〈1 0 0〉 nanowire is less than 1.83 nm ×1.83 nm , the wire undergoes a phase transformation from fcc to bct, and the equilibrium strain increases by an order of magnitude. We then applied a uniform uniaxial strain incrementally to 1.2% to the relaxed nanowires in a molecular statics framework. From the simulation results we computed the effective axial Young's modulus and Poisson's ratios of the nanowire as a function of cross-sectional area. We used two approaches to compute the effective elastic moduli, one based on a definition in terms of the strain derivative of the total energy and another in terms of the virial stress often used in atomistic simulations. Both give quantitatively similar results, showing an increase in Young's modulus with a decrease of cross-sectional area in the nanowires that do not undergo a phase transformation. Those that undergo a phase transformation experience an increase of about a factor of three of Young's modulus. The Poisson's ratio of the 〈1 0 0〉 wires that do not undergo a phase transformation show little change with the cross-sectional area. Those wires that undergo a phase transformation experience an increase of about 10% in Poisson's ratio. The 〈1 1 1〉 wires show, with a decrease of cross-sectional area, an increase in one of Poisson's ratios and small change in the other.

On finitely strained magnetorheological elastomers

Abstract: Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. Due to their strong magnetoelastic coupling response MREs are finding an increasing number of engineering applications, thus necessitating appropriate theoretical descriptions which is the objective of this work. Two different continuum formulations for MREs are presented: an Eulerian (current configuration) based approach using the second law of thermodynamics plus the conservation laws method of mechanics and a new, Lagrangian (reference configuration) based formulation based on the unconstrained minimization of a potential energy functional. It is shown that both approaches yield the same governing equations and boundary conditions. Following a discussion of general properties of the free energy function of MREs, we use a particular such function to illustrate the magnetoelastic coupling phenomena in a cylinder subjected to traction or torsion under the presence of external magnetic fields.

Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

Abstract: In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic–plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress–strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress–strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress–strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.

Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics

Abstract: A computational method (CADD) is presented whereby a continuum region containing dislocation defects is coupled to a fully atomistic region. The model is related to previous hybrid models in which continuum finite elements are coupled to a fully atomistic region, with two key advantages: the ability to accomodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly "converting" the atomistic dislocations into discrete dislocations, or vice-versa. The resulting CADD model allows for the study of 2d problems involving large numbers of defects where the system size is too big for fully atomistic simulation, and improves upon existing discrete dislocation techniques by preserving accurate atomistic details of dislocation nucleation and other atomic scale phenomena. Applications to nanoindentation, atomic scale void growth under tensile stress, and fracture are used to validate and demonstrate the capabilities of the model.

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

Abstract: A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.

A quantum-mechanically informed continuum model of hydrogen embrittlement

Abstract: We present a model of hydrogen embrittlement based upon: (i) a cohesive law dependent on impurity coverage that is calculated from first principles; (ii) a stress-assisted diffusion equation with appropriate boundary conditions accounting for the environment; (iii) a static continuum analysis of crack growth including plasticity; and (iv) the Langmuir relation determining the impurity coverage from its bulk concentration. We consider the effect of the following parameters: yield strength, stress intensity factor, hydrogen concentration in the environment, and temperature. The calculations reproduce the following experimental trends: (i) time to initiation and its dependence on yield strength and stress intensity factor; (ii) finite crack jump at initiation; (iii) intermittent crack growth; (iv) stages I and II of crack growth and their dependence on yield strength; (v) the effect of the environmental impurity concentration on the threshold stress intensity factor; and (vi) the effect of temperature on stage II crack velocity in the low-temperature range. In addition, the theoretically and experimentally observed intermittent cracking may be understood as being due to a time lag in the diffusion of hydrogen towards the cohesive zone, since a buildup of hydrogen is necessary in order for the crack to advance. The predictions of the model are in good quantitative agreement with available measurements, suggesting that hydrogen-induced degradation of cohesion is a likely mechanism for hydrogen-assisted cracking.

Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper

Abstract: Nanoscale contact of material surfaces provides an opportunity to explore and better understand the elastic limit and incipient plasticity in crystals. Homogeneous nucleation of a dislocation beneath a nanoindenter is a strain localization event triggered by elastic instability of the perfect crystal at finite strain. The finite element calculation, with a hyperelastic constitutive relation based on an interatomic potential, is employed as an efficient method to characterize such instability. This implementation facilitates the study of dislocation nucleation at length scales that are large compared to atomic dimensions, while remaining faithful to the nonlinear interatomic interactions. An instability criterion based on bifurcation analysis is incorporated into the finite element calculation to predict homogeneous dislocation nucleation. This criterion is superior to that based on the critical resolved shear stress in terms of its accuracy of prediction for both the nucleation site and the slip character of the defect. Finite element calculations of nanoindentation of single crystal copper by a cylindrical indenter and predictions of dislocation nucleation are validated by comparing with direct molecular dynamics simulations governed by the same interatomic potential. Analytic 2D and 3D linear elasticity solutions based on the Stroh formalism are used to benchmark the finite element results. The critical configuration of homogeneous dislocation nucleation under a spherical indenter is quantified with full 3D finite element calculations. The prediction of the nucleation site and slip character is verified by direct molecular dynamics simulations. The critical stress state at the nucleation site obtained from the interatomic potential is in quantitative agreement with ab initio density functional theory calculation.

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Abstract: Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.

A first-principles measure for the twinnability of FCC metals

Abstract: Twinnability is the property describing the ease with which a metal plastically deforms by twinning relative to deforming by dislocation-mediated slip. In this paper a theoretical measure for twinnability in face-centered-cubic (fcc) metals is obtained through homogenization of a recently introduced criterion for deformation twinning (DT) at a crack tip in a single crystal. The DT criterion quantifies the competition between slip and twinning at the crack tip as a function of crack orientation and applied loading. The twinnability of bulk material is obtained by constructing a representative volume element of the material as a polycrystal containing a distribution of microcracks and integrating the DT criterion over all possible grain and microcrack orientations. The resulting integral expression depends weakly on Poisson's ratio and significantly on three interfacial energies: the stacking-fault energy, the unstable-stacking energy and the unstable-twinning energy. All these four quantities can be computed from first principles. The weak dependence on Poisson's ratio is exploited to derive a simple and accurate closed-form approximation for twinnability which clarifies its dependence on the remaining material parameters. To validate the new measure, the twinnability of eight pure fcc metals is computed using parameters obtained from quantum-mechanical tight-binding calculations. The ranking of these materials according to their theoretical twinnability agrees with the available experimental evidence, including the low incidence of DT in Al, and predicts that Pd should twin as easily as Cu.

Constitutive model for stretch-induced softening of the stress?stretch behavior of elastomeric materials

Abstract: Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress–strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress–strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.

A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations

Abstract: A two-dimensional nonlocal version of continuum crystal plasticity theory is proposed, which is based on a statistical-mechanics description of the collective behavior of dislocations coupled to standard small-strain crystal continuum kinematics for single slip. It involves a set of transport equations for the total dislocation density field and for the net-Burgers vector density field, which include a slip system back stress associated to the gradient of the net-Burgers vector density. The theory is applied to the problem of shearing of a two-dimensional composite material with elastic reinforcements in a crystalline matrix. The results are compared to those of discrete dislocation simulations of the same problem. The continuum theory is shown to be able to pick up the distinct dependence on the size of the reinforcing particles for one of the morphologies being studied. Also, its predictions are consistent with the discrete dislocation results during unloading, showing a pronounced Bauschinger effect. None of these features are captured by standard local plasticity theories.

Chemically induced swelling of hydrogels

Abstract: We consider a continuum model for chemically induced volume transitions in hydrogels. Consistent with experimental observations, the model allows for a sharp interface separating swelled and collapsed phases of the underlying polymer network. The polymer chains are treated as a solute with an associated diffusion potential and their concentration is assumed to be discontinuous across the interface. In addition to the standard bulk and interfacial equations imposing force balance and solute balance, the model involves a supplemental interfacial equation imposing configurational force balance. We present a hybrid eXtended-Finite-Element/Level-Set Method for obtaining approximate solutions to the governing equations of the model. As an application, we consider the swelling of a spherical specimen whose boundary is traction-free and is in contact with a reservoir of uniform chemical potential. Our numerical results exhibit good qualitative comparison with experimental observations and predict characteristic swelling times that are proportional to the square of the specimen radius. Our results also suggest several possible synthetic pathways that might be pursued to engineer hydrogels with optimal response times.

The role of friction on sharp indentation

Abstract: Frictional effects on sharp indentation of strain hardening solids are examined in this paper. The results of finite element simulations in a wide range of solids allow us to derive two simplified equations, accounting for the influence of the friction coefficient on hardness. Comparisons between the simulations and instrumented micro-indentation experiments are undertaken to ensure the validity of the former to metallic materials. Quantitative estimates of the role of friction on the development of pileup and sinking-in around the contact boundary are also given in the paper. These results provide a physical insight into the plastic flow features of distinctly different solids brought into contact with sharp indenters. Overall, the investigation shows that the amount of pileup can be used to set the range of validity of the two hardness equations indicated above. Friction has the largest influence on the contact response of solids exhibiting considerable piling-up effects (whose parameter α >1.12 , see text for details), whereas materials developing moderate pileup or sinking-in are less sensitive to friction. Finally, a methodology is devised to assess the influence of the friction coefficient on mechanical properties extracted through indentation experiments.

The effects of shear on delamination in layered materials

Abstract: The effect of transverse shear on delamination in layered, isotropic, linear-elastic materials has been determined. In contrast to the effects of an axial load or a bending moment on the energy-release rate for delamination, the effects of shear depend on the details of the deformation in the crack-tip region. It therefore does not appear to be possible to deduce rigorous expressions for the shear component of the energy-release rate based on steady-state energy arguments or on any type of modified beam theory. The expressions for the shear component of the energy-release rate presented in this work have been obtained using finite-element approaches. By combining these results with earlier expressions for the bending-moment and axial-force components of the energy-release rates, the framework for analyzing delamination in this type of geometry has been extended to the completely general case of any arbitrary loading. The relationship between the effects of shear and other fracture phenomena such as crack-tip rotations, elastic foundations and cohesive zones are discussed in the final sections of this paper.

A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin

Abstract: This study develops a gradient theory of small-deformation viscoplasticity based on: a system of microforces consistent with its peculiar balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that accounts for the Burgers vector through a free energy dependent on curl H p , with Hp the plastic part of the elastic–plastic decomposition of the displacement gradient. The microforce balance and the constitutive equations, restricted by the second law, are shown to be together equivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differential equations. The first of these is an equation for the plastic strain-rate E p in which the stress T plays a basic role; the second, which is independent of T, is an equation for the plastic spin W p . A consequence of this second equation is that the plastic spin vanishes identically when the free energy is independent of curl H p , but not generally otherwise. A formal discussion based on experience with other gradient theories suggests that sufficiently far from boundaries solutions should not differ appreciably from classical solutions, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form. Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditions need be supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a variational formulation of the flow rule is derived. Finally, we sketch a generalization of the theory that allows for isotropic hardening resulting from dissipative constitutive dependences on ∇ E p .

Spherical indentation of freestanding circular thin films in the membrane regime

Abstract: We present theoretical and experimental results to describe the mechanics of indentation of a clamped circular membrane with a frictionless spherical indenter. Analytical expressions and numerical simulations are presented for the relationships between contact radius, finite indentation strains (and stresses), pre-stretch, loads and deflection. These closed-form solutions are contrasted with point-load models that neglect the contact size (i.e. classical Schwerin-type solutions), and lead to important differences in the indentation strain and load–deflection response. The accuracy of these closed form expressions is illustrated by comparisons with detailed numerical results and experiments on thin elastomer films. We show that the closed-form solutions can be used to extract mechanical properties from indentation testing of freestanding films, with important implications for developing new tests on nanoscale films and/or compliant materials such as polymers and biological substances.

Mode I fracture of sheet metal

Abstract: The perceived wisdom about thin sheet fracture is that (i) the crack propagates under mixed mode I & III giving rise to a slant through-thickness fracture profile and (ii) the fracture toughness remains constant at low thickness and eventually decreases with increasing thickness. In the present study, fracture tests performed on thin DENT plates of various thicknesses made of stainless steel, mild steel, 6082-O and NS4 aluminium alloys, brass, bronze, lead, and zinc systematically exhibit (i) mode I "bath-tub", i.e. "cup & cup", fracture profiles with limited shear lips and significant localized necking (more than 50% thickness reduction), (ii) a fracture toughness that linearly increases with increasing thickness (in the range of 0.5-5 mm). The different contributions to the work expended during fracture of these materials are separated based on dimensional considerations. The paper emphasises the two parts of the work spent in the fracture process zone: the necking work and the "fracture" work. Experiments show that, as expected, the work of necking per unit area linearly increases with thickness. For a typical thickness of I mm, both fracture and necking contributions have the same order of magnitude in most of the metals investigated. A model is developed in order to independently evaluate the work of necking, which successfully predicts the experimental values. further-more, it enables the fracture energy to be derived from tests performed with only one specimen thickness. In a second modelling step, the work of fracture is computed using an enhanced void growth model valid in the quasi plane stress regime. The fracture energy varies linearly with the yield stress and void spacing and is a strong function of the hardening exponent and initial void volume fraction. The coupling of the two models allows the relative contributions of necking versus fracture to be quantified with respect to (i) the two length scales involved in this problem, i.e. the void spacing and the plate thickness, and (ii) the flow properties of the material. Each term can dominate depending on the properties of the material which explains the different behaviours reported in the literature about thin plate fracture toughness and its dependence with thickness. (C) 2003 Elsevier Ltd. All rights reserved.

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Abstract: Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall–Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall–Petch slope turns negative in the nano-range. It is also found that, when the Hall–Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.

Numerical simulation of elasto-plastic deformation of composites: evolution of stress microfields and implications for homogenization models

Abstract: The deformation of a composite made up of a random and homogeneous dispersion of elastic spheres in an elasto-plastic matrix was simulated by the finite element analysis of three-dimensional multiparticle cubic cells with periodic boundary conditions. “Exact” results (to a few percent) in tension and shear were determined by averaging 12 stress–strain curves obtained from cells containing 30 spheres, and they were compared with the predictions of secant homogenization models. In addition, the numerical simulations supplied detailed information of the stress microfields, which was used to ascertain the accuracy and the limitations of the homogenization models to include the nonlinear deformation of the matrix. It was found that secant approximations based on the volume-averaged second-order moment of the matrix stress tensor, combined with a highly accurate linear homogenization model, provided excellent predictions of the composite response when the matrix strain hardening rate was high. This was not the case, however, in composites which exhibited marked plastic strain localization in the matrix. The analysis of the evolution of the matrix stresses revealed that better predictions of the composite behavior can be obtained with new homogenization models which capture the essential differences in the stress carried by the elastic and plastic regions in the matrix at the onset of plastic deformation.

An atomistic-based continuum theory for carbon nanotubes: Analysis of fracture nucleation

Abstract: Carbon nanotubes (CNTs) display unique properties and have many potential applications. Prior theoretical studies on CNTs are based on atomistic models such as empirical potential molecular dynamics (MD), tight-binding methods, or first-principles calculations. Here we develop an atomistic-based continuum theory for CNTs. The interatomic potential is directly incorporated into the continuum analysis through constitutive models. Such an approach involves no additional parameter fitting beyond those introduced in the interatomic potential. The atomistic-based continuum theory is then applied to study fracture nucleation in CNTs by modelling it as a bifurcation problem. The results agree well with the MD simulations.

On the relevance of Continuum Damage Mechanics as applied to the Mullins effect in elastomers

Abstract: The present paper reports and rationalizes the use of Continuum Damage Mechanics (CDM) to describe the Mullins effect in elastomers. Thermodynamics of rubber-like hyperelastic materials with isotropic damage is considered. Since it is demonstrated that stress-softening is not strictly speaking a damage phenomenon, the limitations of the CDM approach are highlighted. Moreover, connections with two-network-based constitutive models proposed by other authors are exhibited through the choice of both the damage criterion and the measure of deformation. Experimental data are used to establish the evolution equation of the stress-softening variable, and the choice of the maximum deformation endured previously by the material as the damage criterion is revealed as questionable. Nevertheless, the present model agrees qualitatively well with experiments except to reproduce the strain-hardening phenomenon that takes place as reloading paths rejoin the primary loading path. Finally, the numerical implementation highlights the influence of loading paths on material response and thereby demonstrates the importance of considering the Mullins effect in industrial design.

The role of binder mobility in spontaneous adhesive contact and implications for cell adhesion

Abstract: The standard view of mechanical adhesive contact is as a competition between a reduction in free energy when surfaces with bonding potential come into contact and an increase in free energy due to elastic deformation that is required to make these surfaces conform. An equilibrium state is defined by an incremental balance between these effects, akin to the Griffith crack growth criterion. In the case of adhesion of biological cells, the molecules that tend to form surface-to-surface bonds are confined to the cell wall but they are mobile within the wall, adding a new phenomenon of direct relevance to adhesive contact. In this article, the process of adhesive contact of an initially curved elastic plate to a flat surface is studied for the case in which the binders that account for adhesion are able to migrate within the plate. This is done by including entropic free energy of the binder distribution in the total free energy of the system. By adopting a constitutive assumption that binders migrate at a speed proportional to the local gradient in chemical potential, the transient growth of an adhesion zone due to binder transport is analyzed. For the case of a plate of very large extent, the problem can be solved in closed form, whereas numerical methods are invoked for the case of a plate of limited extent. Results are presented on the rate of growth of an adhesion zone in terms of system parameters, on the evolution of the distribution of binders and, in the case of a plate of limited extent, on the long-term limiting size of the adhesion zone.

Large electrostrictive actuation of barium titanate single crystals

Abstract: An experimental investigation of the electromechanical behavior of single crystals of the ferroelectric perovskite barium titanate is presented. An experimental setup has been designed to investigate large strain actuation in single crystal ferroelectrics subjected to combined electrical and mechanical loading. Experiments have been performed on initially single domain crystals of barium titanate with (1 0 0) and (0 0 1) orientation at compressive stresses between 0 and 5 MPa. Global strain and polarization histories have been recorded. The electrostrictive response is shown to be highly dependent on the level of applied stress with a maximum strain of 0.9% measured at a compressive stress of about 2 MPa and electric 8eld of about 10 kV/cm. This level of strain is about 5 times higher than in typical commercial piezoelectric PZT. Polarized light microscopy has been used to observe the evolution of the domain pattern simultaneously with the strain and polarization measurement. The observations reveal that the observed large strain behavior is the result of 90° domain switching.