Simulation of springback

Springback, the elastically-driven change of shape of a part after forming, has been simulated with 2-D and 3-D finite element modeling. Simulations using solid and shell elements have been compared with draw-bend measurements presented in a companion paper. Plane-stress and plane-strain simulations revealed the dramatic role of numerical tolerances and procedures on the results. For example, up to 51 integration points through the sheet thickness were required for accuracy within 1%, compared with 5–9 typically acceptable for forming simulations. Improvements were also needed in the number of elements in contact with the tools, and in the numerical tolerance for satisfying equilibrium at each step. Significant plastic straining took place in some cases upon unloading; however the choice of elastic–plastic unloading scheme had little effect on the results. While 2-D simulations showed good agreement with experiments under some test conditions, springback discrepancies of hundreds of percent were noted for one alloy with sheet tension near the yield stress. 3-D simulations provided much better agreement, the major source of error being identified as the presence of persistent anticlastic curvature. Most of the remaining deviation in results can be attributed to inaccuracies of the material model. In particular, the presence of a Bauschinger effect changes the results markedly, and taking it into account provided good agreement. Shell elements were adequate to predict springback accurately for R/t greater than 5 or 6, while solid elements were required for higher curvatures. As R/t approaches 2, springback simulated with solid elements tends to disappear, in agreement with measurements presented in the companion paper and in the literature.     

Role of plastic anisotropy and its evolution on springback

Springback angles and anticlastic curvatures reported for a series of draw-bend tests have been analyzed in detail using a new anisotropic hardening model, four common sheet metal yield functions, and finite element procedures developed for this problem. A common lot of 6022-T4 aluminum alloy was used for all testing in order to reduce material variation. The new anisotropic hardening model extends existing mixed kinematic/isotropic and nonlinear kinematic formulations. It replicates three principal characteristics observed in uniaxial tension/compression test reversals: a transient region with low yield stress and high strain hardening, and a permanent offset of the flow stress at large subsequent strains. This hardening model was implemented in ABAQUS in conjunction with four yield functions: von Mises, Hill quadratic, Barlat three-parameter, and Barlat 1996. The simulated springback angle depended intimately on both hardening law after the strain reversal and on the plastic anisotropy. The springback angle at low back forces was controlled by the hardening law, while at higher back forces the anticlastic curvature, which depends principally on yield surface shape, controlled the springback angle. Simulations utilizing Barlat's 1996 yield function showed remarkable agreement with all measurements, in contrast to simulations with the other three yield functions.     

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.     

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.     

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

In the conventional finite element method (FEM), the dynamic characteristics of a longitudinally vibrating rod with mass density ρ_r, Young's modulus E_r, cross-sectional area A_r and total length ℓ_r are considered to be the same as those of a helical spring with stiffness constant k_r=A_rE_r/ℓ_r and total mass m_r=ρ_rA_rℓ_r. For a lumped-mass model, the mass matrix of a rod element is a 2×2 diagonal one with each of its non-zero coefficients to be equal to one half of the total rod mass (i.e., 0.5m_r). Furthermore, the dynamic characteristics of a rod on the basis of last “lumped-mass” model have been found to be very close to those on the basis of “consistent-mass” model. Thus, one can easily take into account of the inertial effect of a helical spring using a massless one with “one half of its total mass”, respectively, concentrated at its two ends (in Method 2) instead of modeling it by an elastic rod with uniform mass per unit length (in Method 1). When one more spring-mass system is attached to the beam, the total number of unknown constants increases “one” in Method 2 and “two” in Method 1, thus, Method 2 will reduce more effort than Method 1 for studying the dynamic behaviors of a beam carrying a number of spring-mass systems with mass of each helical spring considered. In this paper, the formulations of Methods 1 and 2 are presented first and then the numerical examples are illustrated to confirm the reliability of the presented theory and the developed computer programs. Finally, the effect concerning mass of each helical spring of the spring-mass systems is studied.     

Buckling of thin cylindrical shells: an attempt to resolve a paradox

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t^1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference.Our present principal aim is to explain the observed t^1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point.We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t^2.5, other things being equal.This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t^2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t^1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight.We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there.Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation.     

Influence of glass substrate thickness in laser scribing of glass

The thickness of the glass substrate used in liquid crystal displays continues to be decreased from its original thickness of 1.1 mm for the purpose reducing size and weight. The aim of this study was to clarify the influence of the glass substrate thickness during laser scribing with crack propagation caused by laser heating followed by quick quenching. The laser scribe conditions for soda-lime glass substrates with thickness equal to or less than 1.1 mm were obtained in laser irradiation experiments. Two-dimensional thermal elasticity analysis was conducted with a finite element method based on the scribable conditions obtained in the experiment. The laser scribable conditions can then be estimated by the upper limit of the maximum surface temperature, T_max, and the lower limit of the maximum tensile stress, σ_tmax, in the cooling area, regardless of the glass substrate thickness. There is a substrate thickness with which the maximum tensile stress σ_tmax becomes the largest under each scribe condition. The substrate thickness with which σ_tmax becomes the largest is obtained at a faster scribe velocity for thinner glass substrate and at slower scribe velocity for thicker glass substrate. Owing to these relations, the crack depth also has almost the same tendency as σ_tmax.     

A new weighting function for solving nonlinear oscillation problems

A new approximation technique is presented for solving nonlinear oscillation problems. A solution of form x = x₀ sin ωt is assumed and put into the differential equation, which leads to an algebraic equation which can be solved for ω in terms of x₀. The new technique is to reduce the algebraic equation to an identity at ωt = π/3. This method is compared with other common approximation techniques and with the exact solution for several nonlinear problems. Overall, the new method works better than any other approximation technique. In addition, it has the very practical advantage of being purely algebraic; no integrals need be evaluated, as in several other methods.     

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.     

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.     

An efficient approach to characterize pseudo-merohedral twins by precession electron diffraction: Application to the LaGaO3 perovskite

Pseudo-merohedral twins are frequently observed in crystals displaying pseudo-symmetry. In these crystals, many [u v w] zone axis electron diffraction patterns are very close and can only be distinguished from intensity considerations. On conventional diffraction patterns (selected-area electron diffraction or microdiffraction), a strong dynamical behaviour averages the diffracted intensities so that only the positions of the reflections on a pattern can be considered. On precession electron diffraction patterns, the diffracted beams display an integrated intensity and a “few-beam” or “systematic row” behaviour prevails which strongly reduces the dynamical interactions. Therefore the diffracted intensity can be taken into account. A procedure based on observation of the weak extra-reflections connected with the pseudo-symmetry is given to identify without ambiguity any zone axis. It is successfully applied to the identification and characterization of {1 2 1} reflection twins present in the LaGaO₃ perovskite.     

New cylinder liner surfaces for low oil consumption

Anticipated emission legislation and reduced fuel consumption are the main driving forces when developing new engines. Optimization of the active surfaces in the piston system is one possible way to meet the above demands. In this study the effects of surface topography and texture direction of the ring/liner contact on oil film thickness and friction were simulated and experimentally tested. “Low wear” results from the experimental wear tests with “glide honed” smooth liner surfaces supported the “low friction” simulation results. In addition a new wear volume sensitive surface roughness parameter, R_ktot, based on the Abbot–Firestone bearing area curve was introduced.     

空间弯管仿真与回弹补偿

回弹是空间弯管成形的主要成形缺陷。为了预测和补偿空间弯管成形中的回弹,通过编写用户幅值子程序（UAMP）,将有限元法应用于空间弯管的成形和回弹分析,提出了一种从仿真结果中提取管材中心线的方法。通过圆柱螺线形弯管成形的仿真计算,建立了管材回弹前后中心轴线曲率、挠率的关系。提出了一种基于求解曲线基本方程的空间弯管模具型面补偿方法,论述了曲线几何补偿的数学依据。通过对曲率和挠率(均为非常值)曲线进行补偿计算和成形仿真,对该方法进行了验证。结果表明该方法能够有效补偿回弹误差,保证弯管精度。     

Assessments of the kinetic and dynamic loads sustained by stationary tools during high rate plastic forming

A systematic method for evaluating the kinetic and dynamic loads sustained by stationary tools (as opposed to moving tools for which methods already exist) during high rate plastic forming is examined and exemplified by examples. It is essentially based on the momentum theorem for continua for incompressible flow, utilizing kinematically admissible velocity fields. In steady state forming processes (such as rolling, wire drawing, etc.), the difference between the active load (imposed or calculated a priori) and the reactive load, is formulated rigorously, whereas for non-steady processes (forging, impact extrusion, etc.) the formulation gives merely an approximation to the dynamic effects on the tools. The resulting velocity-dependent reactions on the tools are given in terms of two nondimensional numbers, namely, the “kinetic head” (u₀²/σ₀) (called the Euler Number) and the “dynamic head” (ú₀L/σ₀), which includes the machine speed (u₀), machine acceleration ( ), material density , yield strength ₀ and a characteristic dimension of the product, L. The same two non-dimensional heads emerged previously from energy-balance consideration in Ref. [1], while approximating dynamic loads on moving tools, hence a consistency is demonstrated. These heads are unavoidably multiplied by geometrical functions, which typify the specific process under consideration and may amplify (or diminish) the intensity of the dynamic effects. The present work is focussed on quantifying, by the above method, the inherent difference between the reactive load sustained by the non-moving tool (say, a die) and the acting load carried by the moving tool (piston, ram, etc.) In particular cases of very slow processes, these loads are equal by static equilibrium. In some practical processes (like rolling) their difference appears to be relatively small, whereas in others (like impact extrusion) it appears extremely large.     

An evaluation of the errors in the yield surface for a rotationally symmetric thin shell due to neglecting transverse normal stresses and shell curvature

An estimate has been made of the errors introduced in the plastic analysis of rotationally symmetric shells by the neglect of transverse normal stress σ_z, and the ratio of thickness to the radii of curvature in comparison to unity. It is shown that the latter effect will generally be negligible and that the more important factor is that due to σ_z. An example of a complete sphere under internal pressure is considered, and the further effect of moving the pressure onto the centre line examined.     

A simple approach to estimate failure surface of polymer and aluminum foams under multiaxial loads

From mechanical point of view, it is required to have a criterion for evaluating the failure of cellular solids (foams) under multiaxial loads. Well-documented experimental results in the literature show foams could fail by several mechanisms, e.g., elastic buckling, plastic yielding, brittle crushing or brittle fracture. In the previous years, both theoretical and phenomenological approaches have been applied to obtain the failure surface of various foams. The purpose of this paper is to present a simple approach to estimate the complete failure surface of “non-textured” foams. The predicted results of polymer and aluminum foams are compared with the experimental results reported in the literature. It is found that three selected tests will be sufficient to estimate the complete failure surface of a foam. The recommended testing stress states are σ₁=σ₂=σ₃>0, σ₁=σ₂=σ₃<0, and σ₁=−σ₂=−σ₃ (or σ₁=−σ₂, σ₃=0).     

The influence of complex surface geometry on contact damage in curved brittle coatings

The effects of complex geometry on contact damage in bi-layer systems composed of curved brittle coating layers on compliant polymeric substrate is investigated. Previous studies of this problem utilise relatively simple flat or singly curved (domed) model structures. It is not known the extent to which conclusions driven from such observations may extended to more complex (practical) geometry. Glass plates of 1000 μm thick are used as representative of the brittle coating layer, and epoxy filler under layer as representative of under layer support. A series of doubly curved specimens (having curvatures of 4 and 8 mm) are produced to allow investigation of the influence of complex curvature on the evolution of damage. The specimens are tested by indentation with spheres of 4 mm radius loaded along the convex axis of symmetry. For comparison, some specimens loaded parallel to the axis of symmetry but off-centre. The study explores the influence of supporting geometries on the conditions to initiate and propagate subsurface “radial” cracks, which are believed to be responsible for catastrophic failure of brittle-coating-based structures in certain applications, such as dental crowns. It is demonstrated that critical loads for initiation of radial cracks and the subsequent crack propagation are insensitive to complex geometry, so that simple monotonic indentation “axis and/or off-axis loading” with minimum geometrical complication “flat, simply curved” remains an appropriate route to study the evolution of radial cracks in practical brittle coating structures.     

Reference stresses and temperatures for cylinders and spheres under internal pressure with a steady heat flow in the radial direction

The paper examines the creep behavior of thick cylinders and spheres subjected to internal pressure and a negative temperature gradient in the radial direction. It is found that at stationary state the rate of radial displacement of the vessel wall is simply proportional to the material creep behavior associated with a single stress and temperature. Such “reference stresses” and “reference temperatures” are defined for spheres and cylinders of varying wall thicknesses. These reference stresses and reference temperatures are valid for any creep problem where the material behavior may be characterized by a function of the form exp (γT)σ^m. The extension of these results to variable pressure and temperature loading cases is discussed.     

Quasi-static axial compression of thin-walled circular aluminium tubes

This paper presents further experimental investigations into axial compression of thin-walled circular tubes, a classical problem studied for several decades. A total of 70 quasi-static tests were conducted on circular 6060 aluminium tubes in the T5, as-received condition. The range of D/t considered was expanded over previous studies to D/t=10–450. Collapse modes were observed for L/D10 and a mode classification chart developed. The average crush force, F_AV, was non-dimensionalised and an empirical formula established as F_AV/M_P=72.3(D/t)^0.32. It was found that test results for both axi-symmetric and non-symmetric modes lie on a single curve. Comprehensive comparisons have been made between existing theories and our test results for F_AV. This has revealed some shortcomings, suggesting that further theoretical work may be required. It was found that the ratio of F_MAX/F_AV increased substantially with an increase in the D/t ratio. The effect of filling aluminium tubes with different density polyurethane foam was also briefly examined.     

高强板成形回弹控制方法研究

高强板的成形回弹问题一直是模具设计生产的难点,回弹问题主要表现在预测、控制及补偿三方面。文章主要针对高强板的回弹控制,概述了回弹产生的原因及影响因素的理论研究情况,从工艺及设计两方面提出了控制高强板回弹的办法,以在设计、生产中控制高强板回弹。