J. Tirosh

Rupture instability in hydroforming deep-drawing process

Abstract The critical fluid-pressure locus above which rupture by tensile instability may occur in the hydroforming deep-drawing process, is formulated and tested. The formulation is based on the classical theory of plasticity (with simple power-law hardening and Mises-Hill normal anisotropic yielding) assuming plane strain tensile failure. Further simplifications, such as assuming constant blank thickness and a constant Coulomb friction coefficient, enable one to account for the coupling effect between the self-adjusted blank curvature and the governing material parameters on rupture. Experiments with copper blanks are aimed to demonstrate that under certain conditions, failure by rupture may be prevented if the path of the working fluid pressure nowhere exceeds the predicted critical-pressure locus path. On the other hand, it is shown that the working fluid pressure should nowhere be lower than a predetermined minimum pressure locus to prevent wrinkles at the rim. Thus a distinct operating zone, lying between the upper and the lower pressure loci, is identified and recommended for practical use.

On suppression of plastic buckling in hydroforming processes

Abstract Critical lateral fluid pressure for suppression of plastic buckling of annular plates during deep drawing by hydroforming process is examined. The theoretical prediction, utilizing the energy method, is accompanied by experiments. The outcome enables to program a fluid pressure path (with respect to the punch travel) which may prevent, to some extent, premature buckling at the flange area of the product. The governing geometrical and material parameters (as plate thickness, drawing ratio, strain hardening exponent, buckling modulus, flange-cup transition curvature, etc.) are all viewed through their effects on the critical fluid pressure at buckling. Hydroforming experiments with blanks made of copper, aluminum, steel and stainless steel with various geometries validate the analysis and demonstrate the technological usefulness of the results.

On tube expansion by internal fluid pressure with additional compressive stress

Abstract The idea that in-plane pure shear can be used for reaching “infinite” forming strains is examined via the bulging process of thin walled tubes. The principal stress ratio of −1 (which represents a pure shear) and other negative ratios were attempted by applying simultaneously internal pressure and axial compression in a synchronized fashion. It turned out that bulging of the tubes with negative stress ratios failed by early buckling of the tubes. On the other extreme, tensile dominated bulging failed by diffused (or localized) necking. An optimized loading path for maximizing the bulging strain between the above opposing trends was experimentally explored, guided by a limit analysis formulation. The experiments were done on Aluminum (Al 5052-0) tubes, using a specially-dedicated machine.

On the hydrodynamic deep-drawing process

Abstract The feasibility of deep-drawing aided by hydrodynamic flow beneath the blank is demonstrated. The analytical background for directing the design of the process is based on the limit analysis in plasticity theory coupled with flow analysis of viscous, non-inertial fluid. It is done for isotropic non-hardening material with inclusion of constant friction coefficient as an additional parameter acting on plastic/rigid interfaces. The main attention is focused on approximating the Limit Drawing Ratio (LDR) via upper and lower bound and comparing the conventional process to the suggested hydrodynamic-aided process. The unique features of the process along with its limitations (geometrical and material-wise) are exhibited. The prominent role of the die curvature with regard to LDR is emphasized. A non-dimensional number combining fluid viscosity, punch speed, material strength and a characteristic length of the process (identified as Sommerfeld Number) appears as a useful measure for the clearance height through which the fluid flows. The study is corroborated with experiments.

Buckling prevention by lateral fluid pressure in deep-drawing

Abstract An instability analysis of flange buckling against lateral fluid pressure in deep drawing is considered. It is intended to explain the experimental fact that relatively low fluid pressure when applied to the flange area can suppress buckling. The analysis is based on the approximate ‘energy method’ with the inclusion of the work against the fluid. The attention is focused on the initiation of the deep drawing process, where buckling (of non-hardening material) is most susceptible. A special apparatus which enables the replacement of a rigid blank-holder by a lateral fluid-pressure was used for testing. A general solution to the critical pressure, above which the deep drawing can be terminated without buckling, is provided. The prediction of the critical pressure and the number of the associated buckling ‘waves’ (wrinkles) agree very well with the experiments. The pertinent geometrical and material variables (as blank thickness, drawing ratio, Young modulus, yield strength, etc.) are grouped in nondimensional form and plotted for various parameters to provide an engineering-type solution for potential users.

High speed deep drawing of hardening and rate sensitive solids with small interfacial friction

Abstract An asymptotic expansion solution for a hardening visco-plastic material undergoing a high rate deep drawing process is derived and utilized. The solution incorporates a wide range of variables, some of them not previously studied in this context, as, for example, the dynamic load induced by the high speed of the operation, the material rate sensitivity index and the frictional constraints along the blank/die interfaces. The behaviour of the suggested solution reconstitutes some observed phenomena, like the loading path of the moving punch, the limiting drawing ratio before rupture intercepts and the onset of blank thinning. Comparisons to available experiments and numerical solutions are made.

Mixed-mode fracture angle and fracture locus of materials subjected to compressive loading

Abstract The incipient fracture angle and fracture loci of prenotched brittle-like material subjected to compressive loading are investigated analytically and experimentally. The analysis of the problem includes parameters whose effects on fracture were pronounced via laboratory tests, namely: notch-tip curvature, subcritical microcracks emanating from the notch and crack closure process. Such considerations, jointly with the well-established fracture criteria in tensile loading (like the critical energy release rate, the critical energy density, J -integral and critical maximum stress used in this work) yielded an associated fracture locus for each criterion. Due to the mixed mode nature of the situation ( K 1 and K 2 ) preevaluation of the fracture angle was instrumental. Data on critical (far-field) compressive load along with measured fracture angles performed on PMMA and Tungsten Carbide specimens are used to depict the most suitable fracture locus and thus to distinguish between the various fracture criteria when extended to fracture under compressive loading. An exact expression for the threshold load for complete closure of 2D elliptical cracks is used to delimit the fracture locus.

The role of fibrous reinforcements well bonded or partially debonded on the transverse strength of composite materials

Abstract The transverse strength of fibrous composite is studied analytically, numerically and experimentally, from fracture mechanics viewpoint. A detailed stress interaction of a single fiber, embedded in an elastic matrix, with a micro-crack situated along or near the interface, performs the basis for the suggested strength analysis. Extension to multi-fiber interaction (by Finite Element) with crack in different location and length is examined. The results quantify the effect of some fundamental parameters (geometrical and material) on the transverse strength. Finally, the strengthening capability of composite having a “soft” adhesive interlayer (surrounding each fiber) is considered and implications are discussed.

Strength behavior of toughened polymers by fibrous (or particulate) elastomers

Abstract The stress interaction between a cylindrical inclusion (elastomeric or rigid) and its surrounding microcracked matrix is considered from a fracture mechanics perspective. This represents conditions which prevail in dilute fibrous composites and serves as a first order approximation for spherical inclusion systems, like toughened polymers. The analysis elucidates the effect of the inclusion size on the anticipated strength of the matrix. It represents conditions which prevail in dilute fibrous composites as well as toughened-modified polymers. Depending largely on the elastic properties of the pair constituents, it is found that in some cases (i.e. when matrix fracture precedes particle cavitation or matrix/fibrous debonding) the composite strength appears to be inclusion size-dependent. Specifically, it is found that when the inclusion is relatively rigid (with respect to the matrix) then an optimal inclusion size exists (scaled by the characteristic length of the intrinsic micro-cracks of the matrix) which maximizes the composite strength. When the inclusion is more compliant then its surrounding matrix, or when structural softening intervenes (like cavitation of the inclusion, debonding from the matrix, or alike) it is found that the smaller the inclusion size the larger is the strength of the composite. The calculations are illustrated with two practical situations: highly compliant inclusion (rubber) in relatively brittle-like matrix (epoxy), and highly brittle-like inclusion (styrene -acrylonitrile copolymer, abbreviated as SAN) in compliant matrix (poly-carbonate, abbreviated as PC). Infinite rigidity (as, for instance, carbon fibers in epoxy) and zero rigidity (as, for instance, post cavitated rubber inclusion) are given as well to illustrate the two extreme limits of current actualities. A source of (possible) error in predicting the composite behavior is believed to be the residual stress that prevails in the matrix caused by thermal expansion mismatch of the two constituents, as arises after the curing process. Such calculations are added here whenever the history of the curing process is available. Various phenomenological aspects of fracture behavior, not explainable hitherto, received admissible reasoning.

The role of die curvature in the performance of deep drawing (hydro-mechanical) processes

Abstract A hydromechanical deep drawing process (which replaces the conventional rigid blank-holder tool with a hydrostatic fluid pressure) is utilized to study the roles played by die curvature, interfacial friction, material hardening, etc. in deep drawing performance. The analytical study is based on limit analysis in plasticity (applying both the upper and the lower bounds simultaneously) with a special emphasis on the geometry of the die profile. The resulting relationships between the various parameters obtained through the bounds are backed by an independent numerical solution using Woos finite difference scheme. The associated experiments, with which the limit analysis is compared, were conducted with aluminium blanks at various die radii and with various holding fluid pressures. The relatively close proximity of the above solutions, in describing the observed behaviour of the process, enables one to draw a few general conclusions about the strength of the limit analysis in describing realistic deep drawing processes. Also potential improvements concerning the choice of die radius of curvature and the blank holding force are indicated.