The stability of a vertical cut

Coulomb's problem of the depth to which a trench can be dug in cohesive soil without the sides falling in is investigated. In the present paper the soil is assumed to be perfectly plastic with yield stress c in shear and unit weight γ; internal friction is ignored. Thus the problem will be treated as one of plane strain with the maximum shear stress criterion of failure. Coulomb's own expression for the depth of cut was the upper-bound value 4c/γ (actually 4(c/γ) cot ς, allowing for internal friction, where and ø is the angle of friction). The best upper bound in the literature is 3·83c/γ, given by a slip-circle analysis; the highest lower bound seems to be 2c/γ. Simple distributions of stress which satisfy both equilibrium and the yield condition for the whole field are shown to give a lower bound of 2√2(c/γ). A partial solution is established for 3·2c/γ.     

Linear free vibration analysis of cross-ply laminated cylindrical helical springs

A linear free vibration analysis of symmetric cross-ply laminated cylindrical helical springs is performed based on the first-order shear deformation theory. Considering the rotary inertia, the shear and axial deformation effects, governing equations of symmetric laminated helical springs made of a linear, homogeneous, and orthotropic material are presented in a straightforward manner based on the classical beam theory. The free vibration equations consisting of 12 scalar ordinary differential equations are solved by the transfer matrix method. The overall transfer matrix of the helix is computed up to any desired accuracy. The soundness of the present results are verified with the reported values which were obtained theoretically and experimentally. After presenting the non-dimensional graphical forms of the free vibrational characteristics of (0°/90°/90°/0°) laminated helical spring made of graphite-epoxy material (AS4/3501-6) with fixed–fixed ends, a non-dimensional parametric study is worked out to examine the effects of the number of active turns, the shear modulus in the 1–2 plane (G₁₂), the ratio of the cylinder diameter to the thickness (D/d), and Young's moduli ratio in 1 and 2 directions (E₁/E₂) on the first six natural frequencies of a uniaxial composite helical spring with clamped-free, clamped-simple, and clamped–clamped ends.     

A universal slip-line model with non-unique solutions for machining with curled chip formation and a restricted contact tool

A universal slip-line model and the corresponding hodograph for two-dimensional machining which can account for chip curl and chip back-flow when machining with a restricted contact tool are presented in this paper. Six major slip-line models previously developed for machining are briefly reviewed. It is shown that all the six models are special cases of the universal slip-line model presented in this paper. Dewhurst and Collins's matrix technique for numerically solving slip-line problems is employed in the mathematical modeling of the universal slip-line field. A key equation is given to determine the shape of the initial slip-line. A non-unique solution for machining processes when using restricted contact tools is obtained. The influence of four major input parameters, i.e. (a) hydrostatic pressure (P_A) at a point on the intersection line of the shear plane and the work surface to be machined; (b) ratio of the frictional shear stress on the tool rake face to the material shear yield stress (τ/k); (c) ratio of the undeformed chip thickness to the length of the tool land (t₁/h); and (d) tool primary rake angle (γ₁), upon five major output parameters, i.e. (a) four slip-line field angles (θ, η₁, η₂, ψ); (b) non-dimensionalized cutting forces (F_c/kt₁w and F_t/kt₁w); (c) chip thickness (t₂); (d) chip up-curl radius (R_u); and (e) chip back-flow angle (η_b), is theoretically established. The issue of the “built-up-edge” produced under certain conditions in machining processes is also studied. It is hoped that the research work of this paper will help in the understanding of the nature and the basic characteristics of machining processes.     

Upper-bound anisotropic yield locus calculations assuming 111-pencil glide

Deformation by 111-pencil glide has been analyzed by an upper-bound model which combines a least-shear analysis and Piehler's maximum virtual work analysis. The least-shear analysis gives exact solutions if three 111 slip systems are active, while the maximum work analysis provides solutions for the case of four active slip systems. Independent checks are used to determine which solution method is appropriate.Computer calculations using this model have been made to determine; (1) the orientation dependence of the Taylor factor for axisymmetric deformation; (2) the yield loci for textured materials having [100], [110] and [111] sheet metals and rotational symmetry; (3) the isotropic yield locus for randomly oriented materials; and (4) flow stresses along critical loading paths for various assumed textures with rotational symmetry. The latter calculations indicate that anisotropic yield loci of textured bcc metals with rotational symmetry are much better approximated by σ_x^a + σ_y^a + R¦σ_x − σ_y¦^a = (R + 1)Y^a where R is the strain ratio and Y is the tensile yield strength with an exponent a = 6 rather than with a = 2 as postulated by Hill. It is not known how well upper-bound calculations like these represent actual yielding behavior.     

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.     

On the anomalous behaviour of anisotropic sheet metals

Using Hill's 1948 criterion [1] for anisotropic yielding and the strain ratio, r, it has been shown that the ratio of the balanced biaxial yield stress, σ_b, to the uniaxial tensile yield stress, σ_u, should be > 1 if r > 1 and < 1 if r < 1. Certain experimental results[2] showed that with commercial-purity aluminium, where r < 1, the ratio of σ_b to σ_u was always > 1 in that study. This was termed anomalous behaviour. Hill has proposed a new criterion[3] that not only appears to provide greater flexibility than does his earlier version but can also encompass anomalous behaviour which the earlier version cannot.Four simplified cases of the 1979 criterion have been proposed[3] and to date only one has been subjected to experimental assessment. However, the goals of those studies were not concerned with anomalous behaviour per se. In this paper, all four cases are analysed to determine the interrelationships of the parameters r and m (exponent in Hill's new criterion) required to encompass anomalous behaviour. It is found that for each of the four cases anomalous behaviour is predicted for a range of (m, r) combinations which are presented graphically in this paper.     

A two-level optimization procedure for material characterization of composites using two symmetric angle-ply beams

A two-level optimization procedure for determining elastic constants E₁, E₂, G₁₂, and ν₁₂ of laminated composite materials using measured axial and lateral strains of two symmetric angle-ply beams with different fiber angles subjected to three-point-bending testing is presented. In the first-level optimization process, the theoretically and experimentally predicted axial and lateral strains of a [(45°/−45°)₆]_s beam are used to construct the strain discrepancy function which is a measure of the sum of the squared differences between the experimental and theoretical predictions of the axial and lateral strains. The identification of the material constants is then formulated as a constrained minimization problem in which the best estimates of shear modulus and Poisson's ratio of the beam are determined to make the strain discrepancy function a global minimum. In the second-level optimization process, shear modulus and Poisson's ratio determined in the first level of optimization are kept constant and Young's moduli of the second angle-ply beam with fiber angles different from 45° are identified by minimizing the strain discrepancy function established at this level of optimization. The suitability of the proposed procedure for material characterization of composite materials has been demonstrated by means of a number of examples.     

An elastic–plastic stress analysis steel-reinforced thermoplastic composite cantilever beam

In the present study an analytical elastic–plastic stress analysis is carried out for a low-density homogeneous polyethylene thermoplastic cantilever beam reinforced by steel fibers. The beam is loaded by a constant single force at its free end. The expansion of the region and the residual stress component of σ_x are determined for 0°, 30°, 45°, 60° and 90° orientation angles. Yielding begins for 0° and 90° orientation angles at the upper and lower surfaces of the beam at the same distances from the free end. However, it starts first at the upper surface for 30° and 45° orientation angles. The elastic–plastic analysis is carried out for both the plastic region which spreads only at the upper surface and the plastic region which spreads at the upper and lower surfaces together. The residual stress components of σ_x and τ_xy are also determined. The intensity of the residual stress component is maximum at the upper and lower surfaces of the beam, but the residual stress component of τ_xy is maximum on or around the x-axis. The beam can be strengthened by using the residual stresses. The distance between the plastically collapsed point and the free end is calculated for the same load in the beam for 0°, 30°, 45°, 60° and 90° orientation angles.     

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.     

A tension-torsion machine for testing yield criteria and stress-strain relationships at T = 78°K

A tension-torsion machine in which aluminium and copper specimens are pulled and twisted while being immersed in liquid nitrogen is described. The incorporated load cell and extensometer offer very accurate readings at T = 78°K.The test materials are especially treated to approach conditions of isotropy and homogeneity. Stress-strain curves in simple tension and pure shear are obtained at T = 292°K and T = 78°K. It can be seen that at 78°K the initial yield surface obeys the von Mises yield criterion.     

Lattice distortions in GaN on sapphire using the CBED–HOLZ technique

The convergent beam electron diffraction (CBED) methodology was developed to investigate the lattice distortions in wurtzite gallium nitride (GaN) from a single zone-axis pattern. The methodology enabled quantitative measurements of lattice distortions (α, β, γ and c) in transmission electron microscope (TEM) specimens of a GaN film grown on (0, 0, 0, 1) sapphire by metal-organic vapour-phase epitaxy. The CBED patterns were obtained at different distances from the GaN/sapphire interface. The results show that GaN is triclinic above the interface with an increased lattice parameter c. At 0.85 μm from the interface, α=90°, β=8905° and γ=11966°. The GaN lattice relaxes steadily back to hexagonal further away from the sapphire substrate. The GaN distortions are mainly confined to the initial stages of growth involving the growth and the coalescence of 3D GaN islands.     

Fretting cracking behaviour on pre-stressed aluminium alloy specimens

A new testing rig has been developed which enables fretting tests on pre-stressed specimens to be carried out. Three aluminium alloys, Al-Li 2091, Al-Cu 2024 and Al-Zn 7075, were used in the tests. The imposed amplitude D ranged from ± 10 to ± 75 μm and normal load F_n from 500 to 1000 N. The static external stress σ_S was set as σ_D/10, σ_D/2 and σ_D (σ_D is the fatigue limit). The tests were carried out with a frequency of 1 or 5 Hz up to 10⁶ cycles. In this paper, analysis of fretting behaviour has been carried out using the fretting map concept. The effect of the axial load (pre-stress) on fretting cracking is emphasized.     

A new method for determining the mechanical stability of lubricating greases

Mechanical stability is of central importance when dealing with the long-term service-length of grease-lubricated roller bearings. Poor stability will lead to consistency degradation of the grease, because of mechanical forces between the rolling parts of the bearing. The result can be leakage of grease through seals, or at worst a total failure of the bearing. The present investigation was initiated because present-day methods for prediction of mechanical stability show weak correlation with real service-length. The aim of the project was to develop a useful alternative. In order to fulfil this, both field tests and laboratory tests were carried out. In the field tests, nine different commercial greases were examined in the wheel bearings of five ore waggons, used for transporting ore by railroad from the Kiruna Mine in northern Sweden to Narvik in northern Norway for shipping to foreign markets. The test ore waggons travelled a distance of about 300,000 km during a period of 3 years. Small samples of greases were taken, on eight different occasions, for consistency testing. After the end of the test period, the damage on the bearings was also studied. In the laboratory tests, new undestroyed greases of the same brand as in the field tests were examined using conventional methods, such as the V2F, the Roll Stability Test and the Grease Worker. Comparisons between the field tests and these laboratory tests indicate poor correlation. In addition to these conventional methods, the relevance of the shear strength of the greases to the prediction of the mechanical stability was also tested. The shear stress τ_L depends on the applied pressure p, thus τ_L=τ₀+γ·p where τ₀ is the shear stress at atmospheric pressure. γ is a property of the lubricant in the same way as viscosity or density. It was found that γ correlates well with the mechanical stability in service. Increased γ values lead to a decrease in the mechanical stability. One reasonable explanation is that high γ values correspond to high shear stresses in the grease, and thus severe conditions for the thickener.     

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.     

Closed-form solution for wedge cutting force through thin metal sheets

A simple kinematic model is developed which describes the main features of the process of the cutting of a plate by a rigid wedge. It is assumed in this model that the plate material curls up into two inclined cylinders as the wedge advances into the plate. This results in membrane stretching up to fracture of the material near the wedge tip, while the “flaps” in the wake of the cut undergo cylindrical bending. Self-consistent, single-term formulas for the indentation force and the energy absorption are arrived at by relating the “far-field” and “near-tip” deformation events through a single geometric parameter, the instantaneous rolling radius. Further analysis of this solution reveals a weak dependence on the wedge angle and a strong dependence on friction coefficient. The final equation for the approximate cutting force over a range of wedge semiangles 10° ≤ θ ≤ 30° and friction coefficients 0.1 ≤ μ ≤ 0.4 is: F = 3.28σ₀(δ_t)^0.2l^0.4t^1.6μ^0.4, which is identical in form and characteristics to the empirical results recently reported by Lu and Calladine [Int. J. Mech. Sci.32, 295–313 (1990)].This analysis is believed to resolve a controversy recently developed in the literature over the interpretation of plate cutting experiments.     

High-resolution large-strain measurement of plastically deformed specimen by Fourier phase correlation

The Fourier phase correlation method is applied to position circular marks one by one printed on the specimen before and after deformation in order to measure the large plastic strain. This method is extremely sensitive to the differences between the profile of a mark and noise. Therefore, it detects marks easily even under non-uniform illumination without any processing such as flattening illumination, noise exclusion and boundary enhancement in image transformation into binary codes. In addition, a new method of moving reference images (MRI) to position the observed marks in the sub-pixel range is proposed. The MRI technique introduces the theoretical resolution 1/n (n: number of divisions of one pixel). The MRI method enables the positions of the deformed marks to be determined with a resolution of ±0.1 pixel (standard deviation σ_p is 0.042 pixel). Strain of the tensile specimen can be measured within an error of ±0.0015 (standard deviation σ_ε=0.00055).     

The high strain rate compression of 7039 aluminium

Aluminium alloy 7039 was compressed at strain rates between 2 × 10³ and 2·5 × 10⁴s⁻¹ using a modified Hopkinson bar. At strain rates between 2 × 10³ and 1·2 × 10⁴s⁻¹ there was a linear relationship between the flow stress and strain-rate with a slope corresponding to a macroscopic viscosity of 2·9 kPa s. At strain rates between 1·2 and 2·5 × 10⁴s⁻¹ there was a levelling out of the flow stress, but the data was too scattered to give a definite trend. Due to the opposing effects of linear work-hardening and adiabatic heating, at strains above 0·15 the specimens work-softened at a rate inversely proportional to the square root of the strain rate. At the higher strains, specimens cracked along the dominant adiabatic shear band formed during the compression.     

Wear-resistance mechanism of an Al–12Si alloy reinforced with aluminosilicate short fibers

The wear behavior of an aluminosilicate (Al₂O₃·SiO₂) short-fiber-reinforced Al–12Si alloy composite and the parent Al–12Si alloy were investigated under dry conditions. The results show that the increased wear resistance of Al₂O₃·SiO₂/Al–12Si can be attributed to the formation of a hardened layer in the sub-surface region where realignment and redistribution of fragmented eutectic phase and fragmented aluminosilicate fibers occur during dry sliding.     

A new methodology for determining the stress state of the plastic region in machining with restricted contact tools

In rigid-plastic slip-line theory, once the geometry of the slip-line field is established, the stress state of the plastic region (including the primary and secondary deformation zones) in restricted contact machining is governed by the hydrostatic pressure P_A (at a point on the intersection line of the shear plane and the work surface to be machined) and the frictional shear stress τ on the tool rake face. Based on the recently established universal slip-line model and a detailed study of six representative machining cases, a new methodology for determining the stress state of the plastic region, i.e. maximum value principle, is presented in this paper. According to this principle, the stress state of the plastic region can be determined by giving both P_A and τ their theoretical maximum permissible values. The theoretical maximum permissible values of P_A and τ can be found by satisfying four mechanical and geometrical constraint conditions under which the universal slip-line model applies. A comprehensive assessment factor is introduced in this paper. It is shown that the three machining parameters investigated in this present study, i.e. cutting force ratio, chip thickness ratio, and chip back-flow angle can be simultaneously considered to form a comprehensive criterion to compare predicted and experimental results. The applicable range of the maximum value principle is also discussed.     

Strengthening caused by power law creep deformation of dispersed particles in an elastic matrix

The interaction energy is approximated between an edge dislocation and a particle deformable by power law creep in an elastic matrix. The stress required to overcome the interaction energy barrier is found to be greater than the Orowan stress, and the dislocation bulges to escape the particle. If the ratio of the shear modulus of the matrix to the viscosity of the particle (μt^m/σ₀) is large, the stress required to climb over the particle is larger than the Orowan stress and the dilocation bulges before it climbs. It is concluded that even if the particle is soft enough to exhibit creep, the strengthening of alloys can be achieved by an Orowan mechanism. The critical resolved shear stress (CRSS) of Cu-B₂O₃, obtained experimentally by Onaka et al. [11], agrees closely with that obtained in our analysis. This supports our analysis that the strength of Cu-B₂O₃ alloy at high temperature may be accounted for by the Orowan mechanism and the attraction between a dislocation and viscous particles. The energy and the force to overcome the energy barrier increases significantly with decrease of m, the strain rate exponent associated with the power law creep particle. It is found through analysis that for m < 1.0 and for certain values of μt^m/σ₀ > 1, the particle repulses the dislocation, while for m = 1.0 and for all values of μt^m/σ₀ > 1, the particle attracts the dislocation, which is the expected interaction between an elastic particle and a dislocation in an elastic matrix.