Slip-line Field Solutions Research Articles

Plastic Rotation Factors of Three-Point Bend and Compact Tension Specimens

Abstract Plastic rotation factors (rp) for three-point bend and compact tension specimens were obtained for a range of normalized crack depths (a/W) from 0.25 to 0.88 and power law hardening materials. The theoretical slip-line field solutions were used for non-strain hardening materials (i.e., n = ∞) and the Kumar-German-Shih finite element plastic displacement solutions for strain hardening materials (i.e., n < ∞) in these rp calculations. For crack depth a/W = 0.50, the factors rp are approximately independent of strain hardening if n > 5 and are in reasonable agreement with those values recommended by ASTM Subcommittee E24.08 for calculation of crack tip opening displacements in these two types of specimens. For n ≤ 5 the error in rp becomes appreciably large.

Slip-line field solutions for three-point notch-bend specimens

The slip-line field solutions for three-point bend specimens are reviewed for both deep and shallow notches. Plastic constraint factors, hydrostatic stress at the notch root and rotation constant to enable the crack upp opening displacement to be determined are given for a wide range of notch geometries. These values can be used in the fracture analysis of low strength metal specimens.

Slip-line fields of indirect type for end extrusion through partly rough square dies

New slip-line field solutions are proposed for end extrusion through square dies with slipping friction at the workpiece-die interfaces. The fields are of indirect type and are analysed by the matrix method. The variation of the mean steady state extrusion pressure with reduction has been computed for these fields. It is shown that some of the relations between field parameters applicable to frictionless extrusion also apply to the present solutions.

A comparison of a viscoplastic finite-element model with slip-line field and upper-bound solutions for non-hardening material subjected to plane strain and axisymmetric extrusion

A study of typical solutions provided by the viscoplastic-penalty finite-element model for steady state extrusion covering a wide range of processing variables from low plane strain to large axisymmetric ratios is presented and a critical comparison made with the results from other techniques. Two different treatments of frictional conditions are examined. Practical features of direct and indirect extrusion are analyzed in terms of the kinematic localizations suggested by the numerical model.

Slip-line field solutions of three-point bend specimens with deep notches

Slip-line field solutions of three-point bend specimens with different notch depths, notch angles and notch root radii are presented. For notch angles less than the critical value the plastic region is localized at the ligament but for larger notch angles the plastic region spreads to the flanks of the notch. It is shown that the critical angle decreases with increasing notch depth and decreasing root radius. Solutions for the constraint factor and maximum hydrostatic stress beneath the notch tip are also obtained as a function of both notch depth and root radius for notch angles above and below the critical value.

Statistical analysis of cleavage fracture ahead of sharp cracks and rounded notches

Mechanisms of initiation and unstable propagation of transgranular cleavage cracks are compared for brittle fracture ahead of sharp cracks and rounded notches, e.g. for fatigue pre-cracks and Charpy V-notches, respectively, in standard toughness specimens. The comparison is made over a range of temperatures, from the lower shelf into the ductile/brittle transition region, for a single phase material containing a known distribution of particles where weakest link statistics can be used to model the onset of catastrophic failure. Using linear and nonlinear elastic solutions for the stress distribution ahead of a sharp crack, and slip-line field solutions, modified for a power hardening material, for the rounded notch, statistical modelling is employed to define the critical dimensions ahead of the crack or notch tip where initial cracking events are most probable. The analysis provides an interpretation of the role of stress gradient in governing microscopic fracture behavior. Predictions are evaluated by comparison with experimental results on the low temperature flow, Charpy V-notch and plane strain fracture toughness behavior of a low carbon mild steel with simple ferrite/grain boundary carbide microstructures.

Indentation by a Rack-shaped Punch

Plane strain indentation of a rack-shaped punch into lead specimens is investigated experimentally and theoretically for the purpose of its application to rough forming of large gears. Metal flow during indentation is examined by the visioplasticity method. Proposed slip line field solutions and an upper bound solution explain well the observed metal flow and the actual load diagrams. It is found that teeth can be formed with a precise pitch by successive indentation using a double teeth punch.

Indentation of a rack-shaped punch.

Plate strain indentation of lead specimens of finite thickness is investigated experimentally and theoretically for the purpose of its application to the rough forming of large gear teeth. The metal flow is examined by visioplasticity during indentation. The proposed slip line field solutions and upper bound solutions explain well the observed metal flow and the actual load diagrams. It is found that the teeth can be formed with a precise pitch by successive indentation using a punch with double teeth.

Slip-line fields for steady-state extrusion through frictionless doubly wedge-shaped dies in plane strain

Slip-line field solutions for steady extrusion through doubly wedge-shaped frictionless die are obtained by assuming a quadrilateral rigid region lying adjacent to the smaller angle wedge surface of the die. Several patterns of the slip-line field are presented. The ranges of the extrusion geometries within which those fields are valid are shown.

Some numerical slipline field solutions for drawing through circular dies

A method is presented for the generation of slipline fields and their hodographs for the problem of plane strain drawing a rigid, perfectly plastic (rpp) material through frictionless circular dies. An optimization scheme is given, using a readily available multivariable optimization routine, which will reliably adjust the shape of an assumed slipline to ensure hodograph convergence. A number of programme runs using different geometries show some interesting oscillatory variations in the computed stress distributions along the die surface and across the exit slipline.

A slip-line field analysis of the rolling contact problem at high loads

An analysis of the steady rolling of a rough rigid cylinder over the surface of a rigid-perfectly plastic solid is presented. An exact slip-line field solution is investigated by using some recently developed techniques. It is shown that the solution is not uniquely defined and infinitely many complete solutions can be constructed for given boundary conditions. The numerical results show interesting variations of the deformation pattern and of rolling resistance at high loads. The limits are predicted to the magnitudes of the applied load and braking torque for which the steady rolling is possible.

A slip-line field solution for plane-strain indentation by an obtuse-angle wedge

A slip-line field solution with lines of stress discontinuity is proposed which is shown to be admissible for obtuse-angled wedges with interfacial friction. The dead metal cap solution which is used conventionally when θ − λ ⩾ π 4 has a lower load than the corresponding non-dead metal cap solution. This result is referred to as Haddow's paradox. Using the new solution it is shown that, at the point of transition, corresponding to θ − λ = π 4 , the load coincides with that of the non-dead metal case. Thus Haddow's paradox is inoperative.

A technique for calculating the continued deformation of a class of slipline field solutions

A numerical procedure is described for calculating the continued plane strain deformations of bodies made of rigid perfectly plastic materials. The technique is illustrated by calculating the continued deformation of a notched bar, with circular notch roots, pulled in tension.

An Analysis of Contact Between a Pair of Surface Asperities During Sliding

An analysis of friction junction formed by a pair of interacting hemispherically shaped surface asperities was carried out. Depending on the maximum geometrical interference of the asperities junction deformation can be elastic or plastic. Elastic junctions were analyzed using Hertz solution. Depending on whether the junction is welded or nonwelded it was assumed that the interfacial shear stress is constant and equal to or less than the bulk shear strength of the weaker material. Solution for the case of plastic junction was approximated by Green’s slip-line field solution.

The Adhesive Transfer of the Slip-Tongue and the Wedge

The adhesive wear process was analyzed microscopically by successive observations in the SEM. Fractographic analysis of the transferred fragment was also conducted. Two modes of adhesive transfer were identified. In the first mode, successive slip occurs on the slip planes within the prow and forms an extrusion called “slip-tongue.” A shear crack subsequently propagates from the root of the slip-tongue. In the second mode, successive plowing takes place at the head of the wedge by accompanying the shear crack growth at its trailing end. In both cases, cracks produce a dimpled fracture surface. It was also found that the fragment transferred from the sharp asperity to the blunt asperity. The slip line field solution was introduced to explain the shape effect on wear.

Closed-Die Forging of Rectangular Billets with Diamond-Shaped Dies

The forging of billets in diamond-shaped closed dies has been investigated. Experiments were conducted at room temperature using Plasticine to simulate the forging of hot-steel. Billets of rectangular cross-section of various height/width ratios were forged between diamond-shaped closed dies of different angles. The effects of die and workpiece geometries and lubrication on die-filling capacity and deformation characteristics, were examined. To observe the metal flow during the process, grid lines were printed on one of the end faces of the billet and flow patterns were determined from a series of photographs of the deformed lines taken at increments of die-stroke. From these deformed patterns deformation modes were able to be suggested. A theoretical study of the metal flow in the initial stages of forging was also conducted. Slip-line field solutions are given to predict the incipient modes of deformation and these are compared with the observed flow patterns.

On a complete solution for frictionless non-steady state extrusion in plane strain

The slip-line field for frictionless non-steady state extrusion can be constructed by a suitable combination of the slip-line fields for plane strain indentation by smooth and rough dies. The stress state at the vertex of the rigid regions within the slip-line field solution is checked in accordance with Hill's criterion. A statically admissible stress field is constructed by the use of Bishop's method in which some lines of stress discontinuity are introduced. A series method for determining the line of stress discontinuity is proposed by combining Ewing's series method and the method of undetermined coefficients in a power series solution of the differential equation. Some examples of the complete solution are given.

A Slip Line Field Solution of the Free Oblique Continuous Cutting Problem in Conditions of Light Friction at Chip-Tool Interface

Lee and Shaffer’s slip line field solution [11] for orthogonal cutting is generalized to the free oblique continuous cutting problem in plane strain. Comparison of the results as predicted by this solution with those of the plane strain modified Merchant model [8] and experimental results is achieved for some key metal cutting parameters. It is shown that in some respect, the plane strain modified Merchant model [8] predicts values of parameters which are closer to experimental results.

Effect of deep wedge-shaped notches of small flank angle on plastic failure

Green's slip-line field for the plane-strain plastic bending of a plate with a wedge-shaped notch of small flank angle is generalized to take into account axial force. Two constraint factors are derived, T 1 due to the axial force and M 1 due to the bending moment, where the moment is taken about the mid-point of the ligament length. These constraint factors are mapped in constraint factor space providing general yielding loci. It is shown that the slip-line field solution becomes inadmissible for certain combinations of axial force and bending moment and Rice's upper bound solution is used under these circumstances.

Geometrically self-similar deformations of a plastic wedge under combined shear and compression loading by a rough flat die

A new slip-line field solution for the shearing of a plastic wedge under geometrically similar conditions is described and its relevance to the formation of wear particles from sheared asperities is discussed. The solution is an example of a recently discovered class of pseudo-steady, plane strain deformations in which a portion of the rigid/plastic material rotates rigidly.