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.     

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.     

A continuum plasticity model for the constitutive and indentation behaviour of foamed metals

A yield surface is proposed that can be fitted to the plastic flow properties of a broad class of solids exhibiting plastic compressibility and different yield points in tension and compression. The yield surface is proposed to describe cellular solids, including foamed metals, and designed to be fitted to three experimental results: (1) the compressive stress–strain response (including densification), (2) the difference between the tensile and compressive yield points and (3) the degree of compressibility of the foam, as measured by the lateral expansion during a uniaxial stress compression test. The model is implemented using finite elements and used to study the effects of plastic compressibility on two problems: the compression of a doubly notched specimen and indentation by a spherical indenter. The model is then fitted to the properties of a typical closed cell aluminum foam and used to study indentation into a dense aluminum face sheet on a foam foundation. The dependence of the indentation load–displacement curve on the relevant material and geometric parameters is determined, and a single load–displacement relation is presented which approximates the behaviour of a wide range of practical designs. These results can be used to design against indentation failure of sandwich panels.     

New method to determine the mechanical properties of heat treated steels

A new numerical approach to indentation problems is developed for hardened materials. A relationship between load, displacement, flow stress and strain hardening exponent of heat treated materials with a hard film, is given.This method is based on the minimisation of the error between the experimental curve (load–displacement of the indenter) and the theoretical curve, function of the mechanical and geometrical properties of the studied materials.Comparison of the numerical results with those experimentally obtained from known materials confirms the interest of the method proposed.     

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.     

Flat-topped conical shell under axial compression

Based on experimental observations of a grid-domed textile composite under axial compression, the large deformation mechanisms of a flat-topped conical shell are identified. Accordingly, both elastic model and rigid-plastic model are proposed to describe the collapse process and predict the load–displacement characteristics. In the rigid-plastic analysis, the energies dissipated in bending along plastic hinge lines and in stretching of the thin-wall segments between the plastic hinge lines are taken into account. Analytical expressions describing the load–displacement and energy–displacement relationships during the large deformation process are derived. Illustrated by typical numerical examples, the effects of apical angle of a flat-topped conical shell on its energy absorption capacity are revealed. The respective strain distributions on the conical shell resulted from bending deformation and membrane deformation are presented. A good agreement is shown between the theoretical predictions and experimental results.     

Reproducing hoop stress–strain behavior for tubular material using lateral compression test

Identification of material properties in the hoop direction, such as stress–strain behavior, is essential in tube hydroforming processes. Conventional tests such as uniaxial tension and compression tests have some drawbacks and limitations. In the current investigations a simple technique to identify the stress–strain behavior in the hoop direction for tubular material is introduced, based on the experimental data obtained from tube lateral compression test. In the proposed technique, an assumed stress–strain curve is used in finite element simulation to predict the load deflection curve of the tube lateral compression. An iterative algorithm is used to compare the calculated and experimental load deflection curves until a good agreement with a percentage deviation less than 4% is obtained. The suggested technique was used to obtain the material properties of Cu–40%Zn brass tube. The predicted stress–strain curve was compared with that obtained from uniaxial compression test. Comparison between the experimental and predicted stress–strain curve showed that the proposed technique is effective in the prediction of the material properties from the tube lateral compression test with percentage deviation less than 1%.     

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%.     

Finite-element simulation of the lateral compression of aluminium tube between rigid plates

The lateral compression of aluminium and clad tubes owing to a large deformation is examined by an incremental elasto-plastic finite-element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specially considered. It is mainly expected to predict the buckling process and load–deflection curves for energy dissipation capacity during the design stage, before trials. The high non-linearity of the process due to geometric changes, the inelastic constitutive behavior, and the deformation-dependent boundary conditions are taken into account in an incremental manner. A static explicit approach to the solution is applied, tangent stiffness matrix equation is solved without iteration and the r_min technique is employed to limit the step size to linear relation. The simulated load–deflection curve agrees with a published experimental result. The predicted geometries of the compressed tube clearly demonstrate the processes of the formation of buckling until unloading. The effects of various parameters of the process, such as elastic modulus, strain hardening exponent, tube thickness, friction coefficient and configurations of the clad tube, on the occurrence of buckling of tube are discussed and interpreted in simulation. The present work may be expected to improve the understanding of the buckling mechanism of lateral compression.     

Experimental and theoretical studies on buckling of thin spherical shells under axial loads

In this paper we present experiments, simulation as well as analysis of the collapse behaviour of thin spherical shells under quasi-static loading. Various aluminium spherical shells with variation in geometrical parameters were manufactured by spinning. Experiments were performed on these shells in a universal testing machine and their load–compression histories were obtained on the machine chart recorder. Three-dimensional numerical simulations were carried out for all the specimens tested under quasi-static loading using ANSYS^®. All the stages of collapse of the shell including non-symmetrical lobe formation were simulated. Material, geometric and contact nonlinearities were incorporated in the analysis. The stress–strain curves of standard samples made from the material were used as input. Piecewise linearity was taken in the plastic region of the material curve. Results thus obtained compared with the experiments well.An analysis was also carried out to study the behaviour of shells under axial compression based on the formation of rolling and stationary plastic hinges. These hinges were also simulated numerically and results match the experiments well.     

A study of the plastic buckling of axially compressed cylindrical shells with a thick-shell theory

A thick shell theory is used to calculate the critical load of plastic buckling of axially compressed cylindrical shells. The buckling equations are derived with the principle of virtual work on the basis of a transverse shear deformable displacement field. The deformation theory of plasticity is used for constitutive equations. To fit the uniaxial stress–strain curve, the Ramberg–Osgood equation is used. In the numerical examples special attention is paid to the dependence of the buckling mode on the ratios of radius to thickness R/h and length to radius L/R. This dependence divides the (R/h,L/R)-plane into simply connected regions each of which corresponds to a buckling mode. These regions form a “buckling mode map”.     

Determination of friction models for metallic die–workpiece interfaces

A two-parameter friction model is used which combines the Coulomb friction model and the friction factor yield stress model. The drawback of this two-parameter model is the complex nature of its calibration. In this paper a new technique is proposed to calibrate the model, which utilizes two testpiece geometries, namely the solid cylindrical compression testpiece and the ring compression testpiece. In addition, a mathematical model is required of the true stress–true strain behaviour of the material, so that finite deformation/finite element techniques can be used to accurately predict the compression behaviour of both testpieces.By a combination of careful experimentation, carried out on aluminium alloy and copper testpieces, and of finite element analyses of the testpieces made using the two-parameter friction model, it has been shown that it is possible to derive the true stress–true strain curve for the workpiece materials; and, to calibrate the friction model. The geometrical changes of all testpieces, carefully measured throughout the tests, for a range of four different friction conditions, dry friction, lubricant, lead metal and nylon, have been predicted with good accuracy using the true stress–true strain constitutive models, the two-parameter friction model, and the finite-element analysis procedures. In this way, the proposed approach has been validated.     

Collapse of pressurized elastoplastic tubular members under lateral loads

The present work examines the collapse of tubular members subjected to lateral (transverse) quasi-static loading in the presence of uniform pressure. In particular, it investigates pressure effects on the ultimate lateral load of tubes and on their energy absorption capacity. External pressure is mainly considered, whereas internal pressure effects are also discussed. Tubes are modeled with shell finite elements, accounting for geometric and material nonlinearities. Relatively thick steel and aluminum tubes (D/t50), which exhibit significant inelastic deformations, are considered. Two-dimensional cases are examined first, where lateral loading is imposed by either two rigid plates or by two opposite radial loads. Three-dimensional cases are also analyzed, where the load is applied either through a pair of opposite wedge-shaped indenters or a single spherical indenter. The results are presented in terms of load–deflection curves for different levels of pressure, and indicate that the presence of pressure has significant effects on tube response. Deformed shapes of tubes are depicted and discussed, and comparison with test data from non-pressurized pipes is conducted. Finally, simplified analytical models are presented for two-dimensional and three-dimensional load configurations, which yield closed-form expressions, compare fairly well with the finite element results and illustrate some important features of tube response in an elegant manner.     

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.     

Plastic flow localization in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity is used to study plastic flow localization in ductile materials. Unlike classical plasticity, the thickness of the shear band in MSG plasticity can be determined analytically from a bifurcation analysis, and the shear band thickness is directly proportional to the intrinsic material length, (μ/σ_Y)²b associated with strain gradients, where μ is the shear modulus, σ_Y is the yield stress, and b is the Burgers vector. The shear band thickness also depends on the softening behavior of the material. The analytical solution of the shear strain rate yields that the maximum shear strain rate inside the shear band is two orders of magnitude higher than that outside, which is a clear indication of plastic flow localization. The limitation of the present model is also discussed.     

Heterogeneous tensile test on elastoplastic metallic sheets: Comparison between FEM simulations and full-field strain measurements

The aim of this work is to propose a non-standard tensile test suitable for the identification of material parameters using full-field strain measurements and finite element analysis. The shape of the sample to be used in this new test must verify three criteria: (i) large heterogeneity of the strain in the gauge area, (ii) large strain-paths diversity and (iii) a good sensitivity of the strain field to the material parameters. After identifying the mechanical parameters of a dual-phase steel sheet using σ–ε and r–α curves, samples of different shapes were studied in order to choose the one that presents the best compromise between the three criteria. The comparison between simulated and measured fields shows a qualitative accordance. Taking into account the difference between these fields in the expression of the cost-function to minimize is expected to improve the quality of the identified material parameters.     

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.     

Study of plowing and friction at the surfaces of plastic deformed metals

Experimental and analytical investigations of plowing and friction were conducted at the surfaces of well-polished lead, aluminum, copper, nickel, molybdenum, and tungsten to study the mechanism of the load/penetration dependency. The experimental tests were performed with a Nano-Indenter XP of MTS and a Scanning Probe Microscope (SPM), Nanoscope IIIa of Digital Instruments. In addition to make indentation and measure the hardness and Young’s modulus, the indenter was used to make scratches at the surface of metals under different normal load while the penetration depth and frictional force encountered during the scratching were recorded. The SPM, operated mostly in the contact mode, was used to examine the scratch profile. Under the test conditions, plastic deformation dominated at the surfaces of the metals. An analytical model was established to express plastically deformed contacts, based on plowing of a conical-shaped indenter with a hemispherical tip at a plastic deformed surface. Penetration depth and scratched volume were calculated, which is in good agreement with experimental observation. The frictional coefficient μ was also calculated with the model, which accounted for plowing as well as the adhesion force between the indenter and surface. Beside fair agreement of experimental data and calculated values on μ under the loads applied, the model indicated a dramatic rise in friction coefficient under very low loads, which was not observed in the tests. The discrepancy was discussed, and it was believed that the dramatic increase in μ is for the calculated μ and may be due to the assumed dominant contribution of adhesion force in actual contact load with decreasing external load, and it appears only the adhesion energy Δγ is significant. The actual adhesion energy Δγ between our diamond indenter and metal surface in our test condition might be smaller than the value used in calculation.     

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.