D.W.A. Rees

Plastic flow in the elliptical bulge test

Abstract A theoretical analysis is presented of plastic flow at the pole of an elliptical bulge as it forms through a die from applying normal pressure to thin metallic sheets. The inherent anisotropy present in as-rolled sheets of three steels and a brass is characterized from in-plane r-values. The latter were found from tensile tests for directions parallel and perpendicular to the direction of rolling. The validity of the theory is appraised from experiments which examine the extent to which it can provide equivalence between pole flow for elliptical bulges, with five different aspect ratios, and the flow in tension. For each material, all tests are equivalent to 20% plastic strain, i.e. the full range of uniform tensile strain. As greater strains (up to e P ⋍ 80% ) are achieved from bulge forming, the flow curves diverge downward with increasing die aspect ratio. This geometrical effect on flow is such that narrower elliptical die apertures require greater maximum pressures to attain instability but appear to flow under lower equivalent stresses. It is suggested that as the strain gradients increase, the assumption that the two principal radii of curvature which encompass the pole and the 1 in. radius on which the spherometer contact is made, become suspect. A correction is made by employing an equivalent gauge length. In practice, a smaller radius would be required to define the pole curvatures more accurately. Alternatively, larger die apertures could be used.

Elastic-Plastic Stresses in Rotating Discs by von Mises and Tresca

The von Mises and Tresca yield criteria are combined with an equilibrium equation to provide the elastic-plastic stress distribution within the discs rotating at high speed. The Tresca criterion provides a closed solution that is traditionally associated with this problem. This paper examines and compares this with an alternative solution from the von Mises criterion. The latter requires that a suitable numerical solution to a govening differential equation is matched to the boundary conditions. For this a Runge-Kutta and a predictor-corrector method are combined to ensure a state of yield and continuity in stress at the interface between the inner core of perfectly plastic material and the outer elastic annulus. A comparison between the two solutions in a hollow disc shows that Tresca advances the elastic-plastic interface further than von Mises for a given speed. Thus, the Tresca prediction to the full y plastic speed is lower. Th e distributions of radial and hoop stress corresponding to the two criteria show only subtle differences within the elastic-plastic speed range for this disc. B y contras t, in a solid disc there is a marked difference in th e radial stress predictions. This alters the e distribution of residual stress and the apparent benefit that can be gained from compressive residual stress when prestressed discs are raised to their operating elastic speeds.

Room-temperature r.f. magnetron sputtered ZnO for electromechanical devices

Abstract Zinc oxide has been deposited using an r.f. magnetron sputtering system, from metallic zinc and ceramic zinc oxide targets, in an atmosphere containing 0 to 100% oxygen. Substrate temperatures from 25 to 450°C have been used and several samples have been annealed in air at 430°C. Corning 7059 glass and stainless-steel wafers are used as substrate materials with prior- and post-deposited aluminium as the contacting material when necessary. Characterization of these films has included scanning electron microscopy, X-ray diffraction, resistivity and dielectric measurements. The high thermal-expansion coefficient of the stainless steel dictated the need for zinc oxide deposited at low temperatures. Good dielectric films of highly oriented zinc oxide are now being produced at room temperature and have been used to drive and pick up from a resonant structure. A theory for the deposition and growth mechanisms occurring at room temperature has been proposed.

An examination of yield surface distortion and translation

SummaryIt is shown that the components of a sixth rank anisotropy tensor are physically significant in representing the distortion observed in both the initial and subsequent yield surface for a polycrystalline material. This feature of anisotropy does not appear in the ellipsoidal surface given by previous theories in which second and fourth rank anisotropy tensors are employed. The number of tensor components, for a series function embodying tensor terms in ascending rank, reduces to a manageable number by the imposition of symmetry and coincidence between the axes of stress and principal orthotropic directions. The identification is made between the yield limits, as found from biaxial stress experiments and tensor components in composite sum form. A one-to-one correspondence is found from a further simplification through the assumption of incompressibility. This is confirmed experimentally for an orthotropic rolled copper and copper alloy sheet.An examination is made of the physical significance of a translation tensor which appears in the subsequent yield function. It is shown that the components of this tensor can be directly identified with those of an internal stress tensor that is a consequence of the heterogenous slipped state in a deformed polycrystal.

Anisotropic hardening theory and the Bauschinger effect

Abstract A review of anisotropic hardening theory is presented with particular reference to the Bauschinger effect in reversed torsion and anisotropic yield loci in σ, τ space associated with plastic shear strain history. The Bauschinger effect is obtained experimentally from a series of torsion tests on En3B steel tubes prestrained to a maximum of 10 per cent plastic shear strain. The effect, measured from the stress in reversed torsion at the proportionality limit, is analysed from the theory. It is shown to be consistent with experimental observations made on the translation and contraction of an initial yield locus, that are in marked contrast to the rigid translation of kinematic hardening rules. The degree of shear prestrain is shown to considerably influence the magnitude of the effect, an observation in full support of a theoretical Bauschinger parameter. The present test data together with existing published data for commercially pure aluminium 1100-F and the aluminium alloy Noral 19 S confirm that the controlling parameter is a scalar coefficient of plastic prestrain. The investigation supports a scalar function that is parabolic in the second invariant of plastic prestrain. The effect of yield point definition is examined and a comparison between theoretical and experimental yield loci is presented.

Sheet orientation and forming limits under diffuse necking

This paper examines the influence of orientation (θ) between the principal stress axes and the material axes when modelling the forming limits for a rolled sheet. Stress and strain transformation equations are coupled with orthotropic plasticity theory1 when the stress ratio remains constant. Three limit strains, two normal and one shear, appear in the material axes at diffuse instability. These strains define a forming limit diagram (FLD) which rotates, with increasing θ, about its pole. Shear strains reach a maximum for θ = 45 ° but are absent for θ = 0 and 90 °. A simpler plane presentation of the FLD appears in axes of major and minor principal plastic strains. This shows that the sensitivity of the FLD to orientation depends upon the r-values. Examples are given of 1. (i) a CR steel whose r-values exceed unity but where r1 ∼- r2, i.e., where only a slight deviation exists between the principal axes of stress and strain and 2. (ii) a 6000 aluminium alloy whose r values are different and less than unity. The observations made provide a mathematical model of the experimental Keeler-Goodwin FLD2,3. The use of thickness strain for one axis of the FLD is also considered for when ultrasonic thickness monitoring is more convenient than surface strain measurement. Fracture strain data from bulge forming, punch indentation, and Ericksen tests fall around the limit lines predicted by a diffuse instability model in the region of most interest to users.

Autofrettage theory and fatigue life of open-ended cylinders

Abstract Two new theories of autofrettage are presented for an open-ended, thick-walled cylinder which can take into account work-hardening in the presence of small plastic strains based upon equivalent uniaxial stress-strain data. They apply to an internally pressurised cylinder where the axial force is not reacted by the cylinder wall, as is the case with the closed-end or plane strain condition, but where a close fitting internal piston or plug is supported separately to contain the pressure. It is shown that solutions to the wall stress distributions depend upon the initial yield condition. Both the Tresca and von Mises flow rules are employed to determine applied and residual stress distributions for given elastic-plastic radii in the wall of the cylinder, assuming purely elastic unloading. Since the Tresca theory does not specifically account for the open-end condition the von Mises theory is taken to be the more realistic, particularly for cylinder diameter ratios (W) in excess of ≈3, where the largest discrepancies between theories arise. It is reasoned, from previous experimental observations, that the greatest benefit autofrettage has on fatigue strength applies to an optimum pressure for which the equivalent residual stress in the bore has not involved reversed yielding. The onset of reversed yielding may be estimated most realistically with an account of the Bauschinger effect as supplied by the rule of kinematic hardening. In order to further appraise these theories they are employed to predict the fatigue life of fully autofrettaged W = 2, Ni-Cr-Mo cylinders. For this purpose the stress intensity factor calibration of Parker and Farrow is used and that of Bowie and Freese is modified to account for the presence of the residual stress. Comparisons made with available experimentally-determined fatigue lives confirm that a non-hardening assumption is acceptable with this cylinder geometry.

A theory of autofrettage with applications to creep and fatigue

Abstract A new theory of autofrettage is proposed for a closed-end cylinder which takes into account work-hardening in the presence of small plastic strains based upon equivalent stress-strain data. Comparisons with experiments on carbon and alloy steels shows that more realistic predictions are possible than with the simplified non-hardening theory. Residual stress distributions are determined for given elastic-plastic radii in the wall of the pressurised cylinder assuming purely elastic unloading. The greatest beneficial effect that autofrettage has on fatigue strength is identified with an optimum pressure for which the equivalent residual stress in the bore does not exceed the reversed yield stress. When the latter is determined from the kinematic hardening rule it is shown that considerable improvement can be expected in the fatigue life of a cylinder subjected to a fluctuating internal pressure. The possible beneficial effect of a residual stress state to creep resistance is also considered. It is postulated that when an equivalent hydrostatic stress state exists at the bore under steady-creep conditions the reductions in equivalent creep rate and the rate of damage accumulation will result in an increased creep life.

Yield functions that account for the effects of initial and subsequent plastic anisotropy

SummaryA descriptive initial yield function is presented from an examination of experimentally determined yield loci, component plastic strain paths and Lodes parameters that indicate either a severe textural anisotropy in a material or a slight departure from the von Mises condition. The transition from the initial function to one that describes a subsequent yield surface which translates with the stress vector is developed and compared with experimental results. Observations on the Bauschinger, Swift and hardening effects in subsequent yield loci, defined at the limit of proportionality, are adequately represented through Zieglers translation law when modified for non-linear work hardening. It is shown that the marked distortion, particularly apparent, in the presence of shear stress, may be represented by considering separately the translation for each quadrant of the initial yield locus.

Instability limits to the forming of sheet metals

Abstract In this paper forming limit diagrams are constructed employing predictions from local and diffuse plastic instability. Comparisons are made with test results from: (i) hydrostatic bulging through circular and elliptical dies; and (ii) uniaxial in-plane tension. The sheet materials investigated include steels and brass in both the initially isotropic and anisotropic conditions. It is shown that the test results follow either a diffuse instability prediction or a maximum pressure prediction in the positive strain quadrant of the forming limit diagram (FLD). A local instability prediction applies to the tension-compression strain quadrant. Analyses are based upon plane reductions to Hills homogenous quadratic potential using: (i) experimental r-values for longitudinal and transverse sheet directions: and (ii) Swifts hardening parameters n and e0 from the circular bulge test. In general, instability theories underestimate the in-plane fracture strains when the latter are found either from an etched grid or from the sheet thickness around the fracture. Instability predictions are to be regarded as a useful lower bound to the forming-limit diagrams of ductile materials. Since FLD construction depends upon the parameters (i) and (ii) above, it is through these that the influences of strain history and rate of straining on the diagram may be quantified.