Predictive model to estimate the stress–strain curves of bulk metals using nanoindentation

Many studies have shown that finite element modeling (FEM) can be used to fit experimental load–displacement data from nanoindentation tests. Most of the experimental data are obtained with sharp indenters. Compared to the spherical case, sharp tips do not directly allow the behavior of tested materials to be deduced because these produce a nominally-constant plastic strain impression. The aim of this work is to construct with FEM an equivalent stress–strain response of a material from a nanoindentation test, done with a pyramidal indenter. The procedure is based on two equations which link the parameters extracted from the experimental load–displacement curve with material parameters, such as Young's modulus E, yield stress Y₀ and tangent modulus E_T. We have already tested successfully the relations on well-known pure metallic surfaces. However, the load–displacement curve obtained using conical or pyramidal indenters cannot uniquely determine the stress–strain relationship of the indented material. The non-uniqueness of the solution is due to the existence of a characteristic point (ε_c, σ_c); for a given elastic modulus, all bilinear stress–strain curves that exhibit the same true stress σ_c at the specific true strain ε_C lead to the same loading and unloading indentation curve. We show that the true strain ε_c is constant for all tested materials (Fe, Zn, Cu, Ni), with an average value of 4.7% for a conical indenter with a half-included angle θ=70.3°. The ratio σ_c/ε_c is directly related to the elastic modulus of the indented material and the tip geometry.     

Mechanical analysis of the scratching of metals and polymers with conical indenters at moderate and large strains

Scratch test provides a convenient mean to study the surface mechanical properties and the tribological performances of materials. The representative strain of the material in this test increases with the attack angle β of the indenter and so for a conical indenter increases as its apical angle 2θ decreases. But the mechanical analysis of this test by analytic models is very intricate. First we perform a preliminary discussion of the various aspects of the problem by considering the plane strain scratching of materials by wedges. After we present the conditions of the numerical simulations of the scratch test with conical indenters with a three-dimensional (3D) finite element code. These simulations provide the scratch geometry (contact surface, elastic recovery), the plastic strain map and the volume average plastic strain, the scratch hardness and the force ratio, the apparent friction coefficient μ₀=F_t/W. So we compare the behaviour of polymeric and metallic materials in scratch test at low and large strain and relate their difference in scratching resistance to their rheological properties. Polymers develop more higher elastic strains than metals a phenomenon which is characterised at low strain by the yield stress to Young's modulus ratio ε_e=σ_y/E. For θ=70.3° where pure ploughing occurs we study the scratching of elastic perfectly plastic solids with various values of ε_e under zero friction. Some comparisons with the behaviour in indentation are performed and we study the influence of friction in the scratching of workhardened steel with the same cone. At high strain the main rheological difference is the workhardening behaviour: it is described by a power law for metals and an exponential law for polymers. For θ decreasing from 70.3 to 20° we compare the behaviour of a cold worked steel to the behavour of polycarbonate, a thermoplastic polymer: a transition from ploughing to ploughing–cutting occurs only for steel.     

Tribological characterization of environmentally adapted ester based fluids

Fundamental properties of six synthetic ester base fluids, suitable for the formulation of environmentally adapted lubricants, have been investigated. High pressure viscosity data for the test fluids were obtained through experimental measurements with a high pressure Couette rheometer. The temperature, pressure and viscosity data η(p, T) were parameterized against the Roelands pressure–viscosity equation. Thermal conductivity and specific heat capacity data were obtained using a transient hotwire method, and the EHD friction coefficient, γ, was obtained experimentally as well. The results from these measurements are reported, and the correlation between thermal properties, molecular structure, and the fluid rheology parameters, of the test fluids are discussed.     

Identification of yield stress and plastic hardening parameters from a spherical indentation test

Assuming plastic hardening of metals are specified by the stress–strain curve in the form , the material parameters σ₀, k and m are identified from spherical indentation tests by measuring compliance moduli in loading and unloading of the load–penetration curve. The curve P(h_p) is analytically described by a two term expression, each with different exponents. Here, ε_p and h_p denote the plastic strain and permanent penetration. The proposed identification method is illustrated by specific examples including numerical and physical identification tests.     

The use of finite element creep solutions to obtain J integrals for plane strain cracked members

Numerical values are obtained for the path independent J integral and for crack opening displacement δ, for internally and externally cracked tension members under conditions of plane strain. These parameters have been obtained, for an n-power creeping material, from stationary state solutions using the finite element method. Values of J and δ are compared with available results and are shown to be in good agreement.     

Three-parameter approach for elastic–plastic fracture of the semi-elliptical surface crack under tension

Systematic three-dimensional elastic–plastic finite element analyses are carried out for a semi-elliptical surface crack in plates under tension. Various aspect ratios (a/c) of three-dimensional fields are analyzed near the semi-elliptical surface crack front. It is shown that the developed J–Q annulus can effectively describe the influence of the in-plane stress parameters as the radial distances (r/(J/σ₀)) are relatively small, while the approach can hardly characterize it very well with the increase of r/(J/σ₀) and strain hardening exponent n. In order to characterize the important stress parameters well, such as the equivalent stress σ_e, the hydrostatic stress σ_m and the stress triaxiality R_σ, the three-parameter J–Q_T–T_z approach is proposed based on the numerical analysis as well as a critical discussion on the previous studies. By introducing the out-of-plane stress constraint factor T_z and the Q_T term, which is determined by matching the finite element analysis results, the J–Q_T–T_z solution can predict the corresponding three-dimensional stress state parameters and the equivalent strain effectively in the whole plastic zone. Furthermore, it is exciting to find that the values of J-integral are independent of n under small-scale yielding condition when the stress-free boundary conditions at the side and back surfaces of the plate have negligible effect on the stress state along the crack front, and the normalized J tends to a same value when φ equals about 31.5° for different a/c and n. Finally, the empirical formula of T_z and the stress components are provided to predict the stress state parameters effectively.     

Theoretical and experimental results of the plastic and strain-hardening behaviour of En8 at a controlled rate

This investigation contains two theoretical analyses of the plastic behaviour and work-hardening characteristics of medium carbon steel (En8). The first of these analyses employs Perzyna's visco-plastic constitutive law for strain-rate sensitive and work-hardening material behaviour, and the second uses the strain-rate independent theory of Prandtl and Reuss. Both theories gave good agreement with the time histories of the axial stress, the plastic strains and the stress trajectory. However, these theories gave an under-estimate of the shear stress decay versus time. It is also evident from the present study that neither of these theories could account for the lack of coaxiality between the plastic strain-rate and deviatoric stress vectors observed in the experimental results for the present bilinear deformation path.The objective of the present study was to test the ability of these two models to predict the overall response of a real material to biaxial loading at a controlled rate under a deformation path with an abruptly changing direction. A preliminary comparison was made of the strain-hardening response according to the two models for a material without Lüders strain region. Then existing experimental results of a bilinear deformation path of quasi-static twisting at a reference rate of 10⁻⁶ s⁻¹ beyond the initial yield in torsion (to γ = 10γ₀ where the Lüders strain was exhausted), followed by extension at a constant strain-rate with γ held constant, performed earlier by Meguid and Malvern [1], were used for comparison with these models. It is hoped that this information will guide the designer and the user of stress analyses programs towards more realistic material input data.     

Optimum design of multi-ring composite flywheel rotor using a modified generalized plane strain assumption

A numerical method for calculating the stress and strength ratio distribution of the hybrid rim-type composite flywheel rotor is presented with a consideration of the thermally induced residual stresses. The axisymmetric rotor is divided into several rings and the stiffness matrix for each ring is derived by solving the radial equilibrium equation and the stress–strain–temperature relations. The ring stiffness matrices are assembled into a symmetric global matrix satisfying the continuity equations at each interface with the assumptions of a modified generalized plane strain (MGPS). In the MGPS, the z-directional axial strains are assumed to vary linearly along the radial direction; ε_z=ε₀+ε₁r. The conditions that the z-directional force and the circumferential moment resultants vanish are thus used to solve the z-directional axial strains as well as the radial and circumferential strains. After solving the strain distributions, the on-axis stresses and the strength ratios are calculated at each ring. Three-dimensional finite element method (3D FEM) is then used to verify the accuracy of the present method. The results are also compared with those based on the assumption of a plane stress (PSS). In this case, the analysis of MGPS better matches with 3D FEM results than PSS. An optimum design is then performed maximizing total stored energy (TSE) with the thickness of each composite rim as design variables. The optimal design obtained in this study, which considers material sequence, provides a more effective way of maximizing TSE. It is found that the consideration of the residual stress in the design of the hybrid flywheel rotor is crucial. The result of the optimal designs shows that TSE with consideration of ΔT reduces by about 30%.     

Method of determination of truncation parameters from measured surface profile

The probability density function of the roughness height of a sliding surface is not always Gaussian like that of a truncated surface caused by running-in or mild wear. Therefore, it is important for obtaining contact pressure or frictional characteristics to estimate the truncation level of the non-Gaussian distribution function. This paper presents a method of determination of the two truncation parameters in the truncation model presented by King et al. [Proceedings of the 4th Leeds–Lyon Symposium on Tribology, MEP, London, 1978, p. 333]. The two truncation parameters p and β can be determined by plotting the values of skewness S_k and kurtosis K obtained from a measured profile of surface roughness on the S_k–K diagram calculated with the truncation model for various given values of parameters p and β. The height distributions reproduced by the truncation model with the truncation parameters p and β identified by the present method is in good agreement with the original ones of the measured surfaces.     

Molecular dynamics simulation of deformation behavior in amorphous polymer: nucleation of chain entanglements and network structure under uniaxial tension

In order to clarify the mechanical behavior of molecular chains in amorphous polymers, a molecular dynamics simulation is conducted on a nanoscopic specimen of amorphous polyethylene under uniaxial tension. The specimen involves 3542 random coil molecular chains composed of 500–1500 methylene monomers with about two million methylene groups. The stress–strain curve shows a linear elastic relationship at the initial stage of _zz0.03 at . Then the material “yields” by elongating without stress increase up to the strain of 1.5, where strain hardening appears. Careful investigation of changes in dihedral angle and morphology of all molecular chains reveals that the gauche→trans transition takes place during yielding, generating a new network-like structure composed of entangled molecular clusters and oriented chains bridging them. The strain hardening is due to the directional orientation and stretching of molecular chains between entanglements in the nucleated structure.     

Effect of deformation-dependent material parameters on forming limits of thin sheets

Material properties are deformation history dependent. To take this fact into consideration in forming limit analysis, the material parameters are defined as functions of strain using the Voce equation. These history-dependent material parameters are incorporated in the M–K analysis based on Hill's 1993 yield criterion in which all material parameters are independent, so that the effect of each of these history-dependent parameters on forming limits can be investigated individually. The analysis shows that history-dependent material properties have a significant influence on forming limits. An increasing r-value will increase the limit strain under plane strain (FLD₀), which is different from the traditional M–K analysis. Comparison of predicted results with experimental data illustrates that the consideration of history-dependent material properties can improve forming limit predictions considerably.     

Microstructural tomography of a superalloy using focused ion beam microscopy

A focused ion beam (FIB) microscope has been used to simultaneously depth profile and image the γ–γ^′ microstructure of a nickel base superalloy using normal incidence milling in order to characterize the precipitate microstructure in three dimensions (3D). The normal incidence milling rates of the γ and γ^′ phases in this alloy are closely matched when the orientation of the depth-profiled surface is near , which allows for uniform material removal to depths up to a couple of microns. Depth-profiling experiments consisted of automated ion milling and collection of ion-generated secondary-electron images at specified intervals, and was demonstrated for a voxel resolution of roughly . Image-processing software was used for automated processing of the 2D image sequence to render the γ precipitate structure in 3D.     

Frictionally excited thermoelastic instability in viscoelastic and poroelastic media

Friction materials commonly used in sliding applications, such as clutches and brakes, can be poroelastic and exhibit a viscoelastic behaviour. To the author's knowledge, there are no comprehensive analysis of the influence of poroelastic and viscoelastic material properties on the onset of the phenomenon of frictionally excited thermoelastic instability in sliding systems. This issue is here analysed in some details. Firstly, a linear standard model for the friction material is adopted, introducing an effective complex dynamic modulus E=|E|e^jδ and individuating three independent parameters, E₁, E₂/E₁ and c₂/E₁, that fully describe its viscoelastic behaviour. Subsequently, a similarity between viscoelastic and poroelastic formulation is presented and the three independent parameters introduced are related to the viscosity of the fluid μ_f, the permeability k_p and elastic properties M, α_B of the porous material.The linear elastic formulation proposed by Decuzzi et al. (ASME J. Tribiol. 2001;123:865) has been modified in order to take account of the new constitutive model and the variation of the critical sliding speed with the wave parameter, and viscoelastic/poroelastic properties of the material are examined.It has been found that the susceptibility towards thermoelastic instability increases by increasing both the elastic E₂/E₁ and viscoelastic c₂/E₁ parameters, or by increasing the Biot modulus M and effective stress coefficient α_B, the viscosity μ_f of the fluid, and by reducing the permeability k_p of the porous skeleton. It has been shown that for porous friction materials employed in wet clutches which are weakly viscoelastic, the neglect of its poroelastic behaviour leads to an overestimation of the critical speed smaller than 10%. However, much larger variations are predicted for elastomeric and porous materials with more pronounced viscoelastic behaviour.     

Inverse approach for estimating rheological characteristics of adsorption layer in thin film EHL contacts

A modified Reynolds equation is derived for thin film elastohydrodynamic lubrication (TFEHL) by means of the viscous adsorption theory. This TFEHL theory can be used to explain the deviation between the measured film thickness and that predicted from the convenient elastohydrodynamic lubrication (EHL) theory under very thin film conditions. Results show that the thinner the film, the greater the ratio of the adsorption layer to the total film thickness becomes, and the greater the value of the pressure–viscosity index (z′). An inverse approach is proposed to estimate the pressure distribution based upon the film thickness measurement and to determine the pressure–viscosity index of oil film, and the thickness (δ) and the viscosity ratio (η^*) of the adsorption layer in TFEHL circular contacts. Based on TFEHL theory, the inverse approach can reduce z′ error, and provides a reasonably smooth curve of pressure profile by implementing the measurement error in the film thickness. This algorithm not only estimates the pressure, but also calibrates the film shape. Consequently, it predicts z′, η^*, and δ with very good accuracy. It can also be used to evaluate the lubrication performance from a film thickness map obtained from an optical EHL tester. Results show that the estimated value of z′ is in very good agreement with the experimental data.     

Asymptotic analysis of extrusion of porous metals

Plane strain extrusion of fully dense and porous metals is analysed using asymptotic techniques. The extrusion die is assumed to taper gradually down the extrusion axis. The asymptotic expansions are based on a small parameter ε which is defined as the ratio of the total reduction of the original cross-section to the length of the reduction region. Coulomb's law is used to model the frictional forces that develop along the metal-die interface and the coefficient of friction is assumed to be of order ε. Analytical solutions for the first two terms in the expansions are obtained. In the case of the fully dense metals, it is shown that the leading order [O(1)] solution involves “slab flow.” It is also shown that the next term in the expansion of the solution is O(ε²), and this provides a theoretical justification for the use of the so-called “slab methods” of analysis for dies of moderate slope. An asymptotic analysis of the extrusion of porous metals with dilute concentration of voids is also carried out. Gurson's plasticity model is used to describe the constitutive behavior of the material. The leading order solution is the same as that of the fully dense material and the effects of porosity enter as an O(ε) correction. In order to verify the asymptotic solutions developed, detailed finite element calculations are carried out for both the fully dense and the porous material. The asymptotic solutions agree well with the results of the finite element calculations.     

Tool life monitoring during the diamond turning of electroless Ni–P

Diamond tools become severely worn when machining Ni–P plating materials. Tool life monitoring is therefore essential to avoid the deterioration of workpiece quality. In this paper, in order to better detect tool life while the tools are in use, detailed investigations of the cutting force and acoustic emission (AE) measured during diamond turning process have been made. The results of this tool-life testing show that the cutting force and AE supply valuable information on tool failure; the dynamic component of the thrust force fluctuates chaotically when the tool dulls. This phenomenon can be detected using the 1/f^β power spectrum with a spectral exponent of β > 1. On the other hand, the AE amplitude a tends to rise just when chipping occurs on the cutting edge. This feature can be detected by the amplitude distribution spectrum, in which the AE event rate follows the power law a^−m with a scaling exponent of m < 2. Therefore, we can conclude that a spectral exponent of β > 1 and a scaling exponent of m < 2 can be used as the criteria to gauge tool life, because it was observed at the end of the tool life that tool corner had become worn out and that chipping had occurred on the cutting edge.     

Tribological behavior of reciprocating friction drive system under lubricated contact

The dynamic friction and wear behaviors are investigated in reciprocating friction drive system using a 0.45% carbon steel pair. The effects of various operating parameters on the traction force, stick and slip time, and friction modes are examined under the lubricated contacts. Moreover, the critical operating conditions in classifying three friction modes are also established. Results show that the fluid friction induced by the shearing of lubricant dominates the variation of traction force and produces the positive slope γ at the first period of slip in the traction force–relative sliding velocity curve. The γ value decreases at higher driver speed during stick-slip motion due to the thicker fluid film and shear thinning effect. The γ value increases due to the asperity interactions as the friction region is transferred from stick-slip to sticking with normal load from 196 to 980 N. Furthermore, it is also found that the static friction force is independent of stick time for the tangential loading rate ranged from 1.12 to 16.8 s⁻¹. The transition region produces the severest wear under the different driver speeds, but the wear is insensitive to the friction regions and the severe wear only occurs at higher normal load due to the action of Hertzian contact.     

Mechanism and optimum shape of knife edge for metal sealing

The contacting state of knife-edge seals was investigated and a new type of knife edge was developed. The contacting state of knife-edge seals was divided into three types according to the apex width of the knife edge: (a) penetration type, (b) indentation type, and (c) intermediate type. The developed knife edge had a contacting state of penetration type (a). Because of the narrow apex width of the knife edge, the values of P_c/l for the compressive forces per unit length required for sealing were lower than those of other types of knife-edge seals. The contact pressure required for sealing was nearly equal to the Meyer's hardness in the sealing surface layer, regardless of the surface roughness in turning. The optimum shape of the knife edge was of type (a) and had a ridge in a V-shaped cross-section with a plane inclined 30° off normal and the flat area of its apex finished by lapping was about 35 μm wide. The knife edge made of hard material with optimum shape could be utilized in the cases where the sealing materials were copper, carbon steel and stainless steel, and the values of P_c/l were approximately 15–40, 45–110 and 80–190 kN m⁻¹, respectively.     

Pressure-dependence of dielectric relaxation time in poly(propylene glycol) and its application to high-pressure viscosity estimation

Dielectric permittivity and loss of poly(propylene glycol) with different molecular weights (400–3000) and terminal groups (OH and CH₃) have been measured in the frequency range of 100 Hz to 1.5 MHz. Measurements were conducted over the temperature range 202–293 K under atmospheric pressure and 283–320 K under pressure up to 600 MPa. Two relaxation processes, one with strong absorption in the high-frequency region (α-relaxation) and the other a weak process in the low-frequency region (α′-relaxation), were observed for the OH-terminated samples having molecular weights above 2000 and for all the CH₃-terminated samples. Most of the experimental data under high pressure showed a nonlinear decrease in the logarithm of the frequency of maximum dielectric loss with increasing pressure. The pressure-dependence of the dielectric relaxation time of the α-process was analyzed by several models based on the free-volume concept. The regression results of dielectric relaxation time as a function of pressure were applied to the estimation of high-pressure viscosity. The predicted viscosity showed relatively good agreement with viscosity data obtained from a falling-sphere viscometer.     

The surface chemistry of alkyl and arylphosphate vapor phase lubricants on Fe foil

The surface chemistry of tributylphosphate (TBP) and tricresylphosphate (TCP) on a polycrystalline Fe surface was studied using temperature programmed reaction spectroscopy and Auger electron spectroscopy to illustrate some of the initial steps in the reaction mechanisms of alkyl and arylphosphate vapor phase lubricants. During heating, TBP [(C₄H₉O)₃P=O] adsorbed on the Fe surface decomposes via C–O bond scission to give butyl surface intermediates [C₄H₉–] that react via β-hydride elimination to desorb as 1-butene [CH₃CH₂CH=CH₂] and H₂ without appreciable carbon deposition onto the surface. The thermal decomposition of 1-iodobutane [I-C₄H₉] on Fe was observed to proceed via the same β-hydride elimination mechanism. In contrast to tributylphosphate, meta-tricresylphosphate (m-TCP) [(CH₃–C₆H₄O)₃P=O] decomposes on Fe via P–O bond scission to produce methylphenoxy intermediates [CH₃–C₆H₄O–]. During heating to 800 K, methylphenoxy intermediates either desorb as m-cresol [CH₃–C₆H₄–OH] via hydrogenation or decompose further to generate tolyl intermediates [CH₃–C₆H₄–]. Some of the tolyl intermediates desorb as toluene [CH₃–C₆H₅] via hydrogenation but the majority decompose resulting in H₂ and CO desorption and carbon deposition onto the Fe surface. The P–O bond scission mechanism of m-TCP was verified by showing that the temperature programmed reaction spectra of m-cresol yield products that are almost identical to those of m-TCP. These results provide insight into the origin of the differences in the performance of alkyl and arylphosphates as vapor phase lubricants. The alkylphosphates decompose via alkyl intermediates that readily undergo β-hydride elimination and desorb into the gas phase as olefins, thus removing carbon from the surface. In contrast, the arylphosphates generate aryloxy intermediates by P–O bond scission and aryl intermediates by further C–O bond scission. Neither of these intermediates can undergo β-hydride elimination and thus they decompose to deposit carbon onto the Fe surface. The higher efficiency for carbon deposition may be the primary reason for the superior performance of the arylphosphates over alkylphosphates as vapor phase lubricants.