Showing papers in "Journal of The Mechanics and Physics of Solids in 2003"

Experiments and theory in strain gradient elasticity

Abstract: Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, bh. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, bh, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.

Mechanics of the human red blood cell deformed by optical tweezers

Abstract: The mechanical deformation characteristics of living cells are known to influence strongly their chemical and biological functions and the onset, progression and consequences of a number of human diseases. The mechanics of the human red blood cell (erythrocyte) subjected to large deformation by optical tweezers forms the subject of this paper. Video photography of the cell deformed in a phosphate buffered saline solution at room temperature during the imposition of controlled stretching forces, in the tens to several hundreds picoNewton range, is used to assess experimentally the deformation characteristics. The mechanical responses of the cell during loading and upon release of the optical force are then analysed to extract the elastic properties of the cell membrane by recourse to several different constitutive formulations of the elastic and viscoelastic behavior within the framework of a fully three-dimensional finite element analysis. A parametric study of various geometric, loading and structural factors is also undertaken in order to develop quantitative models for the mechanics of deformation by means of optical tweezers. The outcome of the experimental and computational analyses is then compared with the information available on the mechanical response of the red blood cell from other independent experimental techniques. Potential applications of the optical tweezers method described in this paper to the study of mechanical deformation of living cells under different stress states and in response to the progression of some diseases are also highlighted.

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

Abstract: An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.

A methodology for determining mechanical properties of freestanding thin films and MEMS materials

Abstract: We have developed a novel chip-level membrane deflection experiment particularly suited for the investigation of sub-micron thin films and microelectro-mechanical systems. The experiment consists of loading a fixed–fixed membrane with a line load applied at the middle of the span using a nanoindenter. A Mirau microscope interferometer is positioned below the membrane to observe its response in real time. This is accomplished through a micromachined wafer containing a window that exposes the bottom surface of the specimen. A combined atomic force microscope/nanoindenter incorporates the interferometer to allow continuous monitoring of the membrane deflection during both loading and unloading. As the nanoindenter engages and deflects the sample downward, fringes are formed and acquired by means of a CCD camera. Digital monochromatic images are obtained and stored at periodic intervals of time to map the strain field. Stresses and strains are computed independently without recourse to mathematical assumptions or numerical calibrations. Additionally, no restrictions on the material behavior are imposed in the interpretation of the data. In fact, inelastic mechanisms including strain gradient plasticity, piezo and shape memory effects can be characterized by this technique. The test methodology, data acquisition and reduction are illustrated by investigating the response of 1-μm thick gold membranes. A Young's modulus of 53 GPa , a yield stress of 55 MPa and a residual stress of 12 MPa are consistently measured. The post-yield behavior leading to fracture exhibits typical statistical variations associated to plasticity and microcrack initiation.

Study of particle rearrangement during powder compaction by the Discrete Element Method

Abstract: This paper presents simulations of cold isostatic and closed die compaction of powders based on the Discrete Element Method. Due to the particulate nature of powders, densification of the compact proceeds both through the plastic deformation at the particle contact and the mutual rearrangement of particles. The relative weight of each mechanism on the macroscopic deformation process depends on the contact law, the relative density, and the type of stress exerted on the particles (shear or pressure). 3D computer simulations have been carried out to investigate the role of these parameters on the deformation mechanisms of powder compacts. The effect of rearrangement is studied by comparing simulations that use a homogeneous strain field solution for which local rearrangement is omitted and simulations that include local rearrangement. It is shown that local rearrangement has some effect on average quantities such as the average coordination number, the average contact area and the macroscopic stress. The effect on averaged quantities is much stronger for closed die compaction than for isostatic compaction. However the main effect of local rearrangement is to widen the distribution of the parameters that define the contact (contact area in particular). The results of these simulations are compared to available experimental data and to statistical models that use a homogeneous strain field assumption.

Nanoindentation and the indentation size effect: Kinetics of deformation and strain gradient plasticity

Abstract: A study of the indentation size effect (ISE) in aluminum and alpha brass is presented. The study employs rate effects to examine the fundamental mechanisms responsible for the ISE. These rate effects are characterized in terms of the rate sensitivity of the hardness, ∂ H/ ∂ ln e eff , where H is the hardness and e eff is an effective strain rate in the plastic volume beneath the indenter. ∂ H/ ∂ ln e eff can be measured using indentation creep, load relaxation, or rate change experiments. The activation volume V ∗ , calculated based on ∂ H/ ∂ ln e eff which can traditionally be used to compare rate sensitivity data from a hardness test to conventional uniaxial testing, is calculated. Using materials with different stacking fault energy and specimens with different levels of work hardening, we demonstrate how increasing the dislocation density affects V ∗ ; these effects may be taken as a kinetic signature of dislocation strengthening mechanisms. We noticed both H and ∂ H/ ∂ ln e eff (V ∗ ) exhibit an ISE. The course of V ∗ vs. H as a result of the ISE is consistent with the course of testing specimens with different level of work hardening. This result was observed in both materials. This suggests that a dislocation mechanism is responsible for the ISE. When the results are fitted to a strain gradient plasticity model, the data at deep indents (microhardness and large nanoindentation) exhibit a straight-line behavior closely identical to literature data. However, for shallow indents (nanoindentation data), the slope of the line severely changes, decreasing by a factor of 10, resulting in a “bilinear behavior”.

Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation

Abstract: Nanoindentation experiments have shown that microstructural inhomogeneities across the surface of gold thin films lead to position-dependent nanoindentation behavior [Phys. Rev. B (2002), to be submitted]. The rationale for such behavior was based on the availability of dislocation sources at the grain boundary for initiating plasticity. In order to verify or refute this theory, a computational approach has been pursued. Here, a simulation study of the initial stages of indentation using the embedded atom method (EAM) is presented. First, the principles of the EAM are given, and a comparison is made between atomistic simulations and continuum models for elastic deformation. Then, the mechanism of dislocation nucleation in single crystalline gold is analyzed, and the effects of elastic anisotropy are considered. Finally, a systematic study of the indentation response in the proximity of a high angle, high sigma (low symmetry) grain boundary is presented; indentation behavior is simulated for varying indenter positions relative to the boundary. The results indicate that high angle grain boundaries are a ready source of dislocations in indentation-induced deformation.

A Peierls criterion for the onset of deformation twinning at a crack tip

Abstract: A criterion for the onset of deformation twinning (DT) is derived within the Peierls framework for dislocation emission from a crack tip due to Rice (J. Mech. Phys. Solids 40(2) (1992) 239). The critical stress intensity factor (SIF) is obtained for nucleation of a two-layer microtwin, which is taken to be a precursor to DT. The nucleation of the microtwin is controlled by the unstable twinning energyγut, a new material parameter identified in the analysis. γut plays the same role for DT as γus, the unstable stacking energy introduced by Rice, plays for dislocation emission. The competition between dislocation emission and DT at the crack tip is quantified by the twinning tendencyT defined as the ratio of the critical SIFs for dislocation nucleation and microtwin formation. DT is predicted when T>1 and dislocation emission when T<1. For the case where the external loading is proportional to a single load parameter, T is proportional to . The predictions of the criterion are compared with atomistic simulations for aluminum of Hai and Tadmor (Acta Mater. 51 (2003) 117) for a number of different crack configurations and loading modes. The criterion is found to be qualitatively exact for all cases, predicting the correct deformation mode and activated slip system. Quantitatively, the accuracy of the predicted nucleation loads varies from 5% to 56%. The sources of error are known and may be reduced by appropriate extensions to the model.

Determination of mechanical properties of carbon nanotubes and vertically aligned carbon nanotube forests using nanoindentation

Abstract: Vertically aligned carbon nanotubes (VACNT) have been a recent subject of intense investigation due to the numerous potential applications of VACNTs ranging from field emission and vacuum microelectronic devices to the creation of super-hydrophobic surfaces and as a source of well defined CNTs. In this paper, a new method to determine the mechanical properties of VACNT and constituent nanotubes using nanoindentation tests is proposed. The study of nanoindentation on a VACNT forest reveals a process whereby nanotubes are consecutively bent during the penetration of the indentor. Therefore, the resistance of a VACNT forest to penetration is due to successive bending of nanotubes as the indentor encounters nanotubes. Using a micro-mechanical model of the indentation process, the effective bending stiffness (EI)eff of constituent nanotubes in the VACNT array is then deduced from nanoindentation force-penetration depth curves. A simple method accounting for the multiwalled structure of multiwall nanotubes is used to interpret the obtained (EI)eff in terms of an effective bending modulus Etb, an effective axial modulus Eta, and a wall modulus Etw of a nanotube. Nanoindentation tests on three VACNT forest samples reveal the effective bending modulus of multiwall carbon nanotubes to be E t b =0.91∼1.24 TPa , and effective axial modulus to be Eta=0.90– 1.23 TPa . These values are in good agreement with tests conducted on isolated MWCNTs. Taking the mechanical wall thickness to be 0.075 nm , the nanotube wall modulus is found to be Etw=4.14– 5.61 TPa , which is in good agreement with predictions from atomic simulations. The use of nanoindentation together with the proposed micromechanical model of the successive bending of nanotubes as the indentor penetrates into the forest is hereby shown to result in a novel approach for determining not only the dependence of the indentation resistance on the key structural features of the forest (CNT diameter, length and areal density), but also provides a measure of the stiffness of the constituent carbon nanotubes. This new technique requires no special treatment of the samples, making it promising to apply this method to a large number of tests to determine the statistical properties of CNTs, and implying the potential use of this method as a quality control measurement in mass production.

Superhard silicon nanospheres

Abstract: Successful deposition and mechanical probing of nearly spherical, defect-free silicon nanospheres has been accomplished. The results show silicon at this length scale to be up to four times harder than bulk silicon. Detailed measurements of plasticity evolution and the corresponding hardening response in normally brittle silicon is possible in these small volumes. Based upon a proposed length scale related to the size of nanospheres in the 20– 50 nm radii range, a prediction of observed hardnesses in the range of 20– 50 GPa is made. The ramifications of this to computational materials science studies on identical volumes are discussed.

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

Abstract: A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress–strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading–reloading cycles are presented.

A comparison of nonlocal continuum and discrete dislocation plasticity predictions

Abstract: Discrete dislocation simulations of two boundary value problems are used as numerical experiments to explore the extent to which the nonlocal crystal plasticity theory of Gurtin (J. Mech. Phys. Solids 50 (2002) 5) can reproduce their predictions. In one problem simple shear of a constrained strip is analyzed, while the other problem concerns a two-dimensional model composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear. In the constrained layer problem, boundary layers develop that give rise to size effects. In the composite problem, the discrete dislocation solutions exhibit composite hardening that depends on the reinforcement morphology, a size dependence of the overall stress–strain response for some morphologies, and a strong Bauschinger effect on unloading. In neither problem are the qualitative features of the discrete dislocation results represented by conventional continuum crystal plasticity. The nonlocal plasticity calculations here reproduce the behavior seen in the discrete dislocation simulations in remarkable detail.

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

Abstract: The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh–Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.

On the determinacy of repetitive structures

Abstract: This paper shows that repetitive, infinite structures cannot be simultaneously statically, and kinematically, determinate.

Slip-line modeling of machining with a rounded-edge tool—Part I: new model and theory

Abstract: The effect of tool edge roundness attracts growing attention from the international machining research community due to ever accelerating applications of precision, super-precision, micro-, and nano-machining technologies in a wide variety of modern industries. A new slip-line model for machining with a rounded-edge tool and its associated hodograph are proposed in this paper. The model consists of 27 slip-line sub-regions, each sub-region having its own physical meaning. It is demonstrated that the model simultaneously takes into account nine effects, such as the shear-zone effect and the size effect, which commonly occur in machining. Eight groups of machining parameters, such as the ploughing (parasitic or non-cutting) force and the chip up-curl radius, can be simultaneously predicted from the model. Furthermore, the model incorporates eight slip-line models previously developed for machining during the last six decades as special cases. An additional special case that involves a parallel-sided shear zone can also be derived from the new model. A mathematical formulation of the model is established based on Dewhurst and Collins's (1973) matrix technique for numerically solving slip-line problems. A purely analytical equation is proposed to predict the thickness of the primary shear zone. This equation is also employed to predict the shear strain-rate in the primary shear zone.

Thermal effects in the superelasticity of crystalline shape-memory materials

Abstract: This paper develops a three-dimensional theory for the superelastic response of single-crystal shape-memory materials. Since energetic considerations play a major role in the phase transformations associated with the superelastic response, we have developed the theory within a framework that accounts for the laws of thermodynamics. We have implemented a special set of constitutive equations resulting from the general theory in a finite-element computer program, and using this program have simulated the superelastic response of a single crystal Ti–Ni shape-memory alloy under both isothermal and thermo-mechanically coupled situations. Both manifestations of superelasticity—stress–strain response at fixed temperature and strain–temperature response at fixed stress—are explored. The single-crystal constitutive-model is also used to discuss the superelastic response of a polycrystalline aggregate with a random initial crystallographic texture. The overall features of the results from the numerical simulations are found to be qualitatively similar to existing experimental results on Ti–Ni.

An experimental investigation of shock wave propagation in periodically layered composites

Abstract: In heterogeneous media, scattering due to interfaces/microstructure between dissimilar materials could play an important role in shock wave dissipation and dispersion. In this work, the influence of interface scattering on finite-amplitude shock waves was experimentally investigated by impacting flyer plates onto periodically layered polycarbonate/6061 aluminum, polycarbonate/304 stainless steel and polycarbonate/glass composites. Experimental results (obtained using velocity interferometer and stress gage) show that these periodically layered composites can support steady structured shock waves. Due to interface scattering, the effective shock viscosity increases with the increase of interface impedance mismatch, and decreases with the increase of interface density (interface area per unit volume) and loading amplitude. For the composites studied here, the strain rate within the shock front is roughly proportional to the square of the shock stress. This indicates that layered composites have much larger shock viscosity due to the interface/microstructure scattering in comparison with the increase of shock strain rate by the fourth power of the shock stress for homogeneous metals. Experimental results also show that due to the scattering effects, shock propagation in the layered composites is dramatically slowed down and the shock speed in composites can be lower than that either of its components.

The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations

Abstract: A systematic study of failure initiation in small-scale specimens has been performed to assess the effect of size scale on “failure properties” by drawing on the classical analysis of elliptically perforated specimens. Limitations imposed by photolithography restricted the minimum radii of curvature of the specimen perforations to one micron. By varying the radius of curvature and the size of the ellipses, the effects of domain size and stress concentration amplitude could be assessed separately to the point where the size of individual grains (∼0.3 μm ) becomes important. The measurements demonstrate a strong influence of the domain size under elevated stress on the “failure strength” of MEMS scale specimens, while the amplitude, or the variation, of the stress concentration factor is less significant. In agreement with probabilistic considerations of failure, the “local failure strength” at the root of a notch clearly increases as the radius of curvature becomes smaller. Accordingly, the statistical scatter also increases with decreasing size of the (super)stressed domain. When the notch radius becomes as small as 1 μm the failure stress increases on average by a factor of two relative to the tension values derived from unnotched specimens. This effect becomes moderate for larger radii of curvature, up to a radius of 8 μm (25 times the grain size), for which the failure stress at the notch tip closely approaches the value of the tensile strength for un-notched tensile configurations. We deduce that standard tests, performed on micron-sized, non-perforated, tension specimens, provide conservative strength values for design purposes. In addition, a Weibull analysis shows for surface-micromachined specimens a dependence of the strength on the specimen length, rather than the surface area or volume, which implies that the sidewall geometry, dimensions and surface conditions can dominate the failure process.

Grain boundary versus transgranular ductile failure

Abstract: The competition between intergranular and intragranular fracture is investigated using a bilayer damage model, which incorporates the relevant microstructural features of aluminium alloys with precipitate free zones (PFZ) nearby the grain boundary. One layer represents the grain behaviour: due to precipitation, it presents a high yield stress and low hardening exponent. The other layer represents the PFZ which has the behaviour of a solid solution: it is much softer but with a much higher strain hardening capacity. In both layers, void growth and coalescence is modelled using an enhanced Gurson-type model incorporating the effects of the void aspect ratio and of the relative void spacing. The effects on the ductility (i) of the flow properties of each zone, (ii) of the relative thickness of the PFZ, and (iii) of the particles spacing and volume fraction in the PFZ are elucidated. Comparisons are made with experimental data. Based on the previous analysis, qualitative understanding of trends in the fracture toughness of aluminium alloys can be gained in order to provide a link with the thermal treatment process. (C) 2003 Elsevier Science Ltd. All rights reserved.

Inhomogeneity effects on the crack driving force in elastic and elastic–plastic materials

Abstract: This article evaluates the effect of material inhomogeneities on the crack-tip driving force in general inhomogeneous bodies and reports results for bimaterial composites. The theoretical model, based on Eshelby material forces, makes no assumptions about the distribution of the inhomogeneities or the constitutive properties of the materials. Inhomogeneities are modeled by making the stored energy have an explicit dependence on the reference coordinates. Then the material inhomogeneity effect on the crack-tip driving force is quantified by the term Cinh, which is the integral of the gradient of the stored energy in the direction of crack growth. The model is demonstrated by two model problems: (i) bimaterial elastic composite using asymptotic solutions and (ii) graded elastic and elastic–plastic compact tension specimen using numerical methods for stress analysis.

Hardness trends in micron scale indentation

Abstract: A continuum theory for elastic-plastic solids that accounts for the size-dependence of strain hardening is employed to analyze trends in the indentation hardness test. Strain gradient plasticity theory incorporates an elevation of flow stress when non-uniform plastic deformations occur at the micron scale. Extensive experimental data exists for size-dependence of indentation hardness in the micron range for conical (pyramidal) indenters, and recent data delineates trends for spherical indenters. Deformation induced by rigid conical and spherical indenters is analyzed in two ways: by exploiting an approximation based on spherically symmetric void expansion and by finite element computations. Trends are presented for hardness as a function of the most important variables in the indentation test, including the size of the indent relative to the material length parameters, the strain hardening exponent, the ratio of initial yield stress to Young's modulus, and the geometry of the indenter. The theory rationalizes seemingly different trends for conical and spherical indenters and accurately simulates hardness data presented recently for iridium, a low yield strain/high hardening material. The dominant role of one of the material length parameters is revealed, and it is suggested that the indentation test may the best means of measuring this parameter.

Dynamic crack deflection and penetration at interfaces in homogeneous materials: experimental studies and model predictions

Abstract: We examine the deflection/penetration behavior of dynamic mode-I cracks propagating at various speeds towards inclined weak planes/interfaces of various strengths in otherwise homogeneous isotropic plates. A dynamic wedge-loading mechanism is used to control the incoming crack speeds, and high-speed photography and dynamic photoelasticity are used to observe, in real-time, the failure mode transition mechanism at the interfaces. Simple dynamic fracture mechanics concepts used in conjunction with a postulated energy criterion are applied to examine the crack deflection/penetration behavior and, for the case of interfacial deflection, to predict the crack tip speed of the deflected crack. It is found that if the interfacial angle and strength are such as to trap an incident dynamic mode-I crack within the interface, a failure mode transition occurs. This transition is characterized by a distinct, observable and predicted speed jump as well as a dramatic crack speed increase as the crack transitions from a purely mode-I crack to an unstable mixed-mode interfacial crack.

A continuum damage model for delaminations in laminated composites

Abstract: Delamination, a typical mode of interfacial damage in laminated composites, has been considered in the context of continuum damage mechanics in this paper. Interfaces where delaminations could occur are introduced between the constituent layers. A simple but appropriate continuum damage representation is proposed. A single scalar damage parameter is employed and the degradation of the interface stiffness is established. Use has been made of the concept of a damage surface to derive the damage evolution law. The damage surface is constructed so that it combines the conventional stress-based and fracture-mechanics-based failure criteria which take account of mode interaction in mixed-mode delamination problems. The damage surface shrinks as damage develops and leads to a softening interfacial constitutive law. By adjusting the shrinkage rate of the damage surface, various interfacial constitutive laws found in the literature can be reproduced. An incremental interfacial constitutive law is also derived for use in damage analysis of laminated composites, which is a non-linear problem in nature. Numerical predictions for problems involving a DCB specimen under pure mode I delamination and mixed-mode delamination in a split beam are in good agreement with available experimental data or analytical solutions. The model has also been applied to the prediction of the failure strength of overlap ply-blocking specimens. The results have been compared with available experimental and alternative theoretical ones and discussed fully.

The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis

Abstract: The construction of effective models for materials that undergo martensitic phase transformations requires usable and accurate functional representations for the free energy density. The general representation of this energy is known to be highly non-convex; it even lacks the property of quasi-convexity. A quasi-convex relaxation, however, does permit one to make certain estimates and powerful conclusions regarding phase transformation. The general expression for the relaxed free energy is however not known in the n-variant case. Analytic solutions are known only for up to 3 variants, whereas cases of practical interests involve 7–13 variants. In this study we examine the n-variant case utilizing relaxation theory and produce a seemingly obvious but very powerful observation regarding a lower bound to the quasi-convex relaxation that makes practical evolutionary computations possible. We also examine in detail the 4-variant case where we explicitly show the relation between three different forms of the free energy of mixing: upper bound by lamination, the Reus lower bound, and a lower estimate of the H -measure bound. A discussion of the bounds and their utility is provided; sample computations are presented for illustrative purposes.

Initiation of fracture at the interface corner of bi-material joints

Abstract: Within the context of linear elasticity, a stress singularity of the form Hr λ −1 may exist at the interface corner of a bi-material joint, where r is the radial distance from the corner, H is the stress intensity factor and λ −1 is the order of the singularity. Recent experimental results in the literature support the use of a critical value of the intensity factor H = H c as a fracture initiation criterion at the interface corner. In this paper, we examine the validity and limitations of this criterion for predicting the onset of fracture in a butt joint consisting of a thin layer of an elastic–plastic adhesive layer sandwiched between two elastic adherends. The evolution of plastic deformation at the corner is determined theoretically and by the finite element method, and the solution is compared with the extent of the elastic singular field. It is shown that H c is a valid fracture parameter if h > B ( H c / σ Y ) 1/(1− λ ) where the non-dimensional constant B =100 for β =0 and B =13 for β = α /4. Here, h is the thickness of the adhesive layer, σ Y is the uniaxial yield stress of the bulk adhesive and ( α , β ) are Dundurs’ parameters (Dundurs, J., J. Appl. Mech. 36 (1969) 650). Experimental results for aluminium/epoxy/aluminium and brass/solder/brass sandwiched joints are used to assess the role of plastic deformation on the validity of the failure criterion.

The mechanical strength of polysilicon films : Part 1. The influence of fabrication governed surface conditions

Abstract: In an effort to explain the considerable variations in measured mechanical strength of polysilicon films doped with phosphorous for use in MEMS applications, the influence of the specimen manufacturing processes on the mechanical properties has been examined in connection with varying exposure to 49% hydrofluoric acid (HF). It was found that surface roughness as characterized by groove formation along grain boundaries depends on the HF release time. Surface undulations and crevasses related to grain structure result thus in reduced fracture strength and, in addition, induce errors into the determination of the effective elastic modulus - especially when the latter is determined from flexure configurations. Extensive exposure to HF results in pervasive material degradation, as evidenced by a transition from transgranular to intergranular fracture, and a correspondingly precipitous drop of the film strength with attendant increase in grain boundary material removal. Short times of exposure to HF can result in delamination of a thin surface layer, which is suffcient to initiate an "early" failure. Longer exposure allows HF permeation into the intergranular domains, degrading the body of the material significantly. On the other hand, tests on material from a different source that has undergone different doping and post-processing demonstrated a suppression of this degradation resulting in film strengths that are higher by a factor of two or more. Thus, consideration of additional influences of doping and electro-chemical phenomena during the HF wet release, in association with silicon-metal contacts, is necessary.

The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading

Abstract: We have examined the problem of the dynamic growth of a single spherical void in an elastic-viscoplastic medium, with a view towards addressing a number of problems that arise during the dynamic failure of metals. Particular attention is paid to inertial, thermal and rate-dependent effects, which have not previously been thoroughly studied in a combined setting. It is shown that the critical stress for unstable growth of the void in the quasistatic case is strongly affected by the thermal softening of the material (in adiabatic calculations). Thermal softening has the effect of lowering the critical stress, and has a stronger influence at high strain hardening exponents. It is shown that the thermally diffusive case for quasistatic void growth in rate-dependent materials is strongly affected by the initial void size, because of the length scale introduced by the thermal diffusion. The effects of inertia are quantified, and it is demonstrated that inertial effects are small in the early stages of void growth and are strongly dependent on the initial size of the void and the rate of loading. Under supercritical loading for the inertial problem, voids of all sizes achieve a constant absolute void growth rate in the long term. Inertia first impedes, but finally promotes dynamic void growth under a subcritical loading. For dynamic void growth, the effect of rate-hardening is to reduce the rate of void growth in comparison to the rate-independent case, and to reduce the final relative void growth achieved.

Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading

Abstract: A new experimental technique for accelerated fatigue crack growth tests was recently developed (Du et al., 2001). The technique, which uses piezoelectric actuators, enables application of cyclic loading at frequencies several orders higher than that by mechanical loading. However, the validity of this technique relies on the equivalence between piezoelectric and mechanical loading. In this paper, the behavior of an interfacial crack between a piezoelectric material and an elastic material under in-plane electric loading is studied. The displacement mismatch along a bonded interface due to electric potential loading on the piezoelectric material is modeled by inserting an array of uniformly distributed dislocations along the interface. By means of Fourier transformation methods, the governing equations are converted to an integral equation, which is then converted to a standard Hilbert problem. A closed form solution for stresses, electric field, and electric displacements along the bonded interface is obtained. The results agree very well with those obtained from numerical simulations. The results show that the closed form solution is accurate not only for far field distributions of stresses and electric variables, but also for the asymptotic distributions near the crack tip. The solution also suggests the likelihood of domain switching in the piezoelectric material near the crack tip, a process that may influence the interfacial fracture resistance.

Slip-line modeling of machining with a rounded-edge tool—Part II: analysis of the size effect and the shear strain-rate

Abstract: Part II of the present study quantitatively analyzes orthogonal metal cutting processes based on the new slip-line model proposed in Part I. The applicable range of the model is illustrated, followed by an explanation of the non-unique nature of the model. It is suggested that the tool edge roundness be comprehensively defined by four variables. Namely: tool edge radius, position of the stagnation point on the tool edge, tool–chip frictional shear stress above the stagnation point on the tool edge, and tool–chip frictional shear stress below the stagnation point on the tool edge. The effects of these four variables on eight groups of machining parameters are investigated. These include (1) cutting force, thrust force, resultant force, and the ratio of cutting force to thrust force; (2) ploughing force; (3) chip up-curl radius; (4) chip thickness; (5) tool–chip contact length; (6) thickness of the primary shear zone; (7) average shear strain in the primary shear zone; and (8) average shear strain-rate in the primary shear zone. The importance of tool edge roundness is further reinforced by a series of new research findings made in this paper. It is revealed that the size effect highly depends on the material constitutive behavior in machining. The dependence of the thickness of the primary shear zone and the dependence of the magnitude of shear strain-rate in the primary shear zone on the tool edge radius are well demonstrated. A surprisingly good agreement between theory and experiments is reached.

The size effect on void growth in ductile materials

Abstract: We have extended the Rice–Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio el/R0, where l is the intrinsic material length on the order of microns, e the remote effective strain, and R0 the void size. For micron-sized voids and small remote effective strain such that el/R0⩽0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that el/R0>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For el/R0>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice–Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.