A generalised CAD/CAM solution to the three-dimensional off-centric extrusion of shaped sections: analysis

In analytical method based on the upper-bound theory is proposed to investigate the three-dimensional off-centric extrusion of arbitrarily shaped sections from arbitrarily shaped billets through linearly converging and smooth curved dies. A set of generalised kinematically admissible velocity fields are derived on the assumption of Bezier-type streamlines that provide compatibility of surface representation with most CAD/CAM systems. To obtain a more realistic non-uniform metal flow, a special velocity function was also incorporated into the derived velocity fields and work hardening effect of the material considered. Based on the method presented, for a given reduction in area, material property, friction condition and off-centric positioning of the exit cross-section, predictions of the deforming grid pattern, curvature of the extruded product as well as upper-bound to the extrusion pressures may be obtained. The above procedure is highlighted and commented upon.     

An analytical formulation as an alternative to FEM software giving strain and stress distributions for the three-dimensional solution of extrusion problems

A new improved analytical method based on the upper bound theorem is presented for the solution of extrusion problems. This method has been formulated in such a way as to eliminate the deficiencies in the previous works in order to enable it as a powerful analytical tool as an alternative to finite element software. The proposed formulation is a general method of solution, which could be used for the analysis of many bulk forming processes. However, in this paper, the forward extrusion of a square section from a round billet has been presented as an example. Kinematically admissible velocity fields were computed which gave a more physically realistic material flow patterns as compared to previous works. The distribution of strain and stress as well as the results for the effect of process parameters on the extrusion load and die geometry was also given. These results were compared with FEM data to observe the accuracy and effectiveness of the present method. For the extrusion of square sections from round billets, detailed analysis of strain distribution on the exit section was carried out using the components of the power due to internal deformation, interface friction, and velocity discontinuities at the entry and exit surfaces. Comparison with experimental data was made, and verification of the theoretical results was carried out. The improvement of the results computed using the present method was shown by comparison with the previous works.     

Formulation of a new generalized kinematically admissible velocity field with a variable axial component for the forward extrusion of shaped sections

For most three-dimensional analytical solutions proposed for the extrusion of shaped sections, the axial component of the velocity vector has been assumed to be constant at each cross section throughout the deforming zone. This shortcoming means that these velocity fields are not in accordance with the reality of the extrusion problem, and hence, the upper bounds based on such fields give high values for the extrusion pressure. To overcome this, a new formulation has been presented in this paper for which a kinematically admissible velocity field has been developed using a variable axial velocity component. For this purpose, curved surfaces of velocity discontinuities at the entry and exit have been proposed and incorporated into the formulation for the extrusion of shaped sections from circular billets. As an example, a square profiled section has been chosen for the extrusion problem. The upper bound on extrusion pressure was computed using the new formulation. It was shown that the initial velocity discontinuity surface at the entry to the deforming region was flat, and as one travels into the deforming region towards the exit section, the velocity discontinuity surface gradually became convex, having the highest convexity at the exit section. This was contrary to what has been suggested in the literature so far. The measure of convexity depends on the extrusion parameters which have been investigated in this work. Experiments were also carried out to verify the theoretical results, and good agreements were observed between the two. Comparison of the present results with similar previous works showed good improvements as well.     

A new formulation for three-dimensional extrusion and its application to extrusion of clover sections

The metal flow in the extrusion process is an important factor in controlling the mechanical property of the extruded products. It is, however, difficult to predict the metal flow in three-dimensional extrusion of complicated sections due to the difficulty in representing the geometry of the die surface and in expressing the corresponding velocity field. In this study a new kinematically admissible velocity field for a generalized three-dimensional flow is derived, in which the flow is bounded by the die surface expressed by an analytic function. Then, by applying the upper-bound method to the derived velocity field, the flow pattern as well as the upper-bound extrusion pressure are obtained. As a computational example, extrusion of clover sections from round billets is chosen. A new method of die surface representation is proposed by which there is a smooth transition of die contour from the die entrance to the die exit. Computation is carried out for work-hardening materials such as aluminium and steel. The analysis takes into account the effect of product shape complexity, lubrication condition and reduction of area on extrusion pressure, average effective strain and distribution of effective strains on the cross-section of the extruded product.     

An exact analytical solution for the self-weight deflection of circular plates of constant thickness

An exact analytical solution, not previously noted in the literature, for the self-weight deflection of a circular plate is obtained by superposition of various elementary solutions due to Love, and the accuracy of the existing approximate procedures is examined. The expressions for stress components are obtained using the axisymmetric stress-strain relations and the validity of the derived solution is justified by satisfying the field equations and the required boundary conditions.     

An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system

The aim of this study is to develop an approximate analytic solution for nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam with an attached heavy intra-span mass. A geometric nonlinearity due to midplane stretching is considered and Newton’s second law of motion along with Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model, are employed to derive the nonlinear equations of motion. The method of multiple timescales is applied directly to the governing equations of motion, and nonlinear natural frequencies and vibration responses of the system are obtained analytically. Regarding the resonance case, the limit-cycle of the response is formulated analytically. A parametric study is conducted in order to highlight the influences of the system parameters. The main objective is to examine how the vibration response of a plain (i.e. without additional adornment) beam is modified by the presence of a heavy mass, attached somewhere along the beam length.     

A generalized method for analysis of three-dimensional extrusion of arbitrarily-shaped sections

Despite increasing demand for and application of three-dimensional extrusion of various sections through continuous dies, so far very little work has been done by systematic and general analysis to predict the plastic flow properly. For effective die design, efficient design method and the related method of theoretical analysis are required for extrusion of complicated sections. In generalized three-dimensional extrusion of sections through continuous dies, a new method of die surface representation, using blending function and Fourier series expansion, is proposed by which smooth transitions of die contour from the die entrance to the die exit are obtained. The flow patterns as well as the upper-bound extrusion pressures are obtained on the basis of the derived velocity field. The effects of area reduction, product shape complexity, die length and frictional condition are discussed in relation to extrusion pressure, the distorted grid pattern and distribution of the final effective strain on the cross-section of the extruded billet. As computational examples for arbitrarily-shaped products rounded rectangles and ellipses are chosen for the extruded sections. Experiments are carried out for aluminum alloys at room temperature for a rounded square section and an elliptic section. In order to visualize the plastic flow, the grid-marking technique is employed. The theoretical predictions both in extrusion load and deformed pattern are in good agreement with the experimental results.     

A new method for bearing design in the metal extrusion of profiled sections

The International Journal of Advanced Manufacturing Technology - Bearing design is a major issue in controlling material flow and producing high-quality extruded products. Adjusting the bearing...     

Upper bound analysis for extrusion at various die land lengths and shaped profiles

The effects of die land lengths, a rarely investigated die extrusion parameter on the die-shaped profiles, on the extrusion pressures are investigated and presented. The analyses of the extrusion pressures by the upper bound method have been extended for the evaluations of the extrusion pressures to complex extruded sections such as square, rectangular, I,- and T-shaped sections with power of deformation due to ironing effect at the die land taken into account. The extrusion pressure contributions due to the die land evaluated theoretically for shaped sections considered are found to increase with die land lengths for any given percentage reduction and also increase with increasing percentage die reductions at any given die land length. The effect of die land lengths on the extrusion pressure increases with increasing complexity of die openings geometry with I-shaped section giving the highest extrusion pressure followed by T-shaped section, rectangular, circular-shaped die openings with square section die opening, giving the least extrusion pressure for any given die reduction at any given die land lengths.     

An analytical solution of a finite bearing with an inclined journal

A solution of Reynolds' equation for a finite bearing with an inclined journal is attempted. A solution (pressure) is expanded as a series of circular trigonometric functions of β′, the coordinate corresponding to circumferential distances, with coefficients as functions of z, the coordinate corresponding to axial distances. Substitution of such a series for pressure in the equation gives an infinite set of coupled ordinary differential equations with varying coefficients. Forms of solutions of this set of equations are assumed in terms of arbitrary constants and with restrictions of continuity at zero values of the inclinations. The arbitrary constants are evaluated using differential equations and boundary conditions. The integrations of pressure for forces and torques are converted into single integrations which are readily evaluated by digital computer.     

侧向挤压陶瓷型芯模具设计与型芯质量评估

断面积较小的型芯在浇铸的时候要经受一定的冲击力，需要有较高的强度，否则会发生严重的断芯现象。为了和目前现有的浇注成形方法的比较，本文利用侧向挤压的方法进行陶瓷型芯的挤压实验，设计了一套侧向挤压模具装置，挤出的产品具有较好的强度，解决了断芯的问题，也证明了该方法的可行性与高效性。     

An analytical solution to heat conduction with a moving heat source

This paper deals with an analytical solution to heat conduction in the medium subjected to a moving heat source. It evaluates the temperature distribution around a rectangular shape source moving at a constant speed along the axis of a bar. The transient temperature field from a moving heat source was analyzed using a Fourier series procedure. The most interesting result of the theory, is the derivation of a single formula capable of predicting the cooling time and cooling rate with a fairly good accuracy for ranges of temperature. Because of the passage of the heat source, the rise of temperature produced at a given near the source, tends to rapidly become constant. Several sample problems are discussed and illustrated, and comparisons with numerical approaches where these can also be used made. The results show that these solutions are in good agreement with the numerical results.     

An analytical solution of the ECAE process of bi-layer strip using a Bezier type streamline

Co-extrusion of strips of different strength and ductility is a process in the production of bimetal rods of desirable physical and mechanical properties. Equal channel angular extrusion (ECAE) is an innovative process for applying severe plastic deformation to materials. The present study is concerned with the ECAE of different bi-layer strips. Developing a kinematically admissible velocity field using a Bezier-type streamline is the aim of present research. In this analysis, using the mentioned velocity field and the upper bound theorem, the extrusion force as well as the strain distribution in the deformation zone of the ECAE process are predicted. The solution takes into account the die profile, friction conditions, strength, arrangement and thickness ratio of the layer materials. Experimental results of ECAE of aluminum/copper and aluminum/mild steel bi-layer strips showed a good agreement with the predicted results.

     

薄膜生产线挤出压力自校正预测控制

挤出机压力控制是薄膜生产线中非常重要的一个环节,挤出压力的稳定与否直接导致薄膜产品的质量.针对实际生产中挤出机压力复杂的非线性特性,提出了一种基于自校正预测的快速控制算法.根据生产线的实际情况,系统进行周期递推参数辨识,最终完成对挤出压力的准确和快速的控制.实际运行结果表明了该方法的有效性和优越性.     

Preparation of cross sections of thin-coated sheet steel for analytical electron microscopy observation

A method of preparing thinned cross sections of coated sheet steel (Galvalume) for observation in an analytical electron microscope (AEM) was developed. Steel panels dipped for 4 sec in baths of 55% by weight (wt pct) Al-1.7 wt pct Si-balance Zn, and 55 wt pct Al-6.88 wt pct Si-balance Zn at 610°C were examined. The ratio of the intensity of the K and L x-ray lines from the same element was used in determining the foil thickness. The AEM results compared favorably to results obtained using electron microprobe analysis (EMPA).     

Analysis of three-dimensional extrusion of sections through curved dies by conformal transformation

A new method of analysis is proposed for the extrusion of arbitrarily shaped sections through curved die profiles. A kinematically admissible velocity field is found by deriving the equation of a stream line. Conformal transformation of a unit circle onto a section is utilized in the derivation. The upper-bound method is then applied to determine the extrusion pressure for the rigid-perfectly plastic material. The redundant work relating to the velocity discontinuities at the entrance and the exit is included in the formulation. The general formulation for an arbitrary cross section is obtained by use of conformal transformation. The upper-bound pressure for extrusion through curved die profiles is computed for a complex section with a curved boundary. Two curved die profiles widely used are chosen to compare the effects of die profiles. From the derived velocity field, the upper-bound extrusion pressures are also computed for the extrusion of regular polygons and rectangles of various aspect ratios. The effects of sectional shape, die profile and interfacial friction at the die surface are discussed.     

Use of pure carbon specimen holders for analytical electron microscopy of thin sections.

The use of a simple specimen holder, made of pure carbon, for analytical electron microscopy of sections of biological material is reported. Interference of the specimen holder with the X-ray spectrum from the specimen is minimized. The contribution of the holder to the continuum spectrum is consistent with theoretical predictions.     

An analytical solution for inhomogeneous strain within cylinders of silicon and effect on quantum band structure under the double-punch test

An analytical solution for inhomogeneous strain distributions within a finite cylinder of silicon under the double-punch test is obtained. The stress function method is employed and a new expression for the stress function is proposed so that all of the governing equations and the boundary conditions are satisfied exactly. The solution for isotropic cylinders under the double-punch test is recovered as a special case. Numerical results show that the strain singularities are usually developed near two end surfaces, but the strain distributions are relatively uniform in the central part of the cylinder. The largest tensile strain is always induced along the axis of loading. In addition, based on the envelope-function method of energy-band theory and quantum mechanics, the effect of external loads on the valence-band structure of silicon is analyzed. The spin–orbit interaction is considered. It is found that external load under the double-punch test can alter considerably the quantum behavior of energy-band structure of silicon, which manifests in the change of the constant-energy surfaces of the heavy-hole band, the light-hole band, the split-off band and the corresponding conductivity masses. The present study provides an alternative method to investigate the electro-optic properties of strained silicon.     

Further investigation into extrusion of trocoidal gear sections considering three-dimensional plastic flow

In the present study, theoretical development proposed in previous work carried out in extrusion of clover sections in relation to a generalized die design method is presented in the extended scope for three-dimensional extrusion of trocoidal gear sections from round billets with experimental verification. Computations are carried out for clover and trocoidal gear sections. The CAD/CAM of the suggested dies is introduced for the experiments. Experiments are carried out for a clover section and a trocoidal gear section with eight teeth. Al 2024 aluminum billets were used as the working material. Half-cut specimens are used for flow visualization of the extrusion process. The theoretical predictions are in good agreement with the experimental results in extrusion load and metal flow.     

General analytical shakedown solution for structures with kinematic hardening materials

The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly.