1. Stress and strain 2. Plasticity 3. Strain hardening 4. Plastic instability 5. Temperature and strain-rate dependence 6. Work balance 7. Slab analysis and friction 8. Friction and lubrication 9. Upper-bound analysis 10. Slip-line field analysis 11. Deformation zone geometry 12. Formability 13. Bending 14. Plastic anisotropy 15. Cupping, redrawing and ironing 16. Forming limit diagrams 17. Stamping 18. Hydroforming 19. Other sheet forming operations 20. Formability tests 21. Sheet metal properties.

Metal Forming: Mechanics and Metallurgy

Recent experimental evidence points to limitations in characterizing the critical strain in ductile fracture solely on the basis of stress triaxiality A second measure of stress state, such as the Lode parameter, is required to discriminate between axisymmetric and shear-dominated stress states This is brought into the sharpest relief by the fact that many structural metals have a fracture strain in shear, at zero stress triaxiality, that can be well below fracture strains under axisymmetric stressing at significantly higher triaxiality Moreover, recent theoretical studies of void growth reveal that triaxiality alone is insufficient to characterize important growth and coalescence features As currently formulated, the Gurson Model of metal plasticity predicts no damage change with strain under zero mean stress, except when voids are nucleated Consequently, the model excludes shear softening due to void distortion and inter-void linking As it stands, the model effectively excludes the possibility of shear localization and fracture under conditions of low triaxiality if void nucleation is not invoked In this paper, an extension of the Gurson model is proposed that incorporates damage growth under low triaxiality straining for shear-dominated states The extension retains the isotropy of the original Gurson Model by making use of the third invariant of stress to distinguish shear dominated states The importance of the extension is illustrated by a study of shear localization over the complete range of applied stress states, clarifying recently reported experimental trends The extension opens the possibility for computational fracture approaches based on the Gurson Model to be extended to shear-dominated failures such as projectile penetration and shear-off phenomena under impulsive loadings

Modification of the Gurson Model for shear failure

In this work, yield surfaces were measured for binary aluminum-magnesium sheet samples which were fabricated by different processing paths to obtain different microstructures. The yielding behavior was measured using biaxial compression tests on cubic specimens made from laminated sheet samples. The yield surfaces were also predicted from a polycrystal model using crystallographic texture data as input and from a phenomenological yield function usually suitable for polycrystalline materials. The experimental yield surfaces were found to be in good agreement with the polycrystal predictions for all materials and with the phenomenological predictions for most materials. However, for samples processed with high cold rolling reduction prior to solution heat treatment, a significant difference was observed between the phenomenological and the experimental yield surfaces in the pure shear region. In this paper, a generalized phenomenological yield description is proposed to account for the behavior of the solute strengthened aluminum alloy sheets studied in this work. It is subsequently shown that this yield function is suitable for the description of the plastic behavior of any aluminum alloy sheet.

Yield function development for aluminum alloy sheets

https://www.tpbin.com/Uploads/Subjects/9d1f5b4a-4c05-4912-8462-d017c42a9268.pdf

Mechanical response and texture evolution of AZ31 alloy at large strains for different strain rates and temperatures

The aim of this paper is to incorporate plastic anisotropy into constitutive equations of porous ductile metals. It is shown that plastic anisotropy of the matrix surrounding the voids in a ductile material could have an influence on both effective stress–strain relation and damage evolution. Two theoretical frameworks are envisageable to study the influence of plastic flow anisotropy: continuum thermodynamics and micromechanics. By going through the Rousselier thermodynamical formulation, one can account for the overall plastic anisotropy, in a very simple manner. However, since this model is based on a weak coupling between plasticity and damage dissipative processes, it does not predict any influence of plastic anisotropy on cavity growth, unless a more suitable choice of the thermodynamical force associated with the damage parameter is made. Micromechanically-based models are then proposed. They consist of extending the famous Gurson model for spherical and cylindrical voids to the case of an orthotropic material. We derive an upper bound of the yield surface of a hollow sphere, or a hollow cylinder, made of a perfectly plastic matrix obeying the Hill criterion. The main findings are related to the so-called ‘scalar effect’ and ‘directional effect’. First, the effect of plastic flow anisotropy on the spherical term of the plastic potential is quantified. This allows a classification of sheet materials with regard to the anisotropy factor h ; this is the scalar effect. A second feature of the model is the plasticity-induced damage anisotropy. This results in directionality of fracture properties (‘directional effect’). The latter is mainly due to the principal Hill coefficients whilst the scalar effect is enhanced by ‘shear’ Hill coefficients. Results are compared to some micromechanical calculations using the finite element method.

Plastic potentials for anisotropic porous solids

Principles of plastic flow theory large strains tensile instability bending membrane analysis of circular shells stretching drawing stretching and drawing steady state forming of cylindrical shells exercises.

Mechanics of sheet metal forming

In this chapter, a simple approach to the analysis of circular shells is given, and deep drawing of circular cups is examined in detail. This can be viewed as two processes; one is stretching sheet over a circular punch, and the other is drawing an annulus inwards. The two operations are connected at the cylindrical cup wall, which is not deforming, but transmits the force between both regions. This overestimates the limiting drawing ratio and investigates the various factors that influence the maximum blank size. Quite large reductions are possible in this process and several ironing dies may be placed in tandem. In such a case, the dies may be spaced widely apart to ensure that the wall has left one die before entering the next. This ensures that the entry stress is zero and prevents excessive wall stresses below the die. In ironing aluminum beverage cans, three dies are used and the wall thickness is reduced to about a third of the original thickness. This increases the cup height greatly.

8 – Cylindrical deep drawing

This chapter shows how sheet may be bulged to an approximately spherical shape by fluid pressure. This process is used to obtain mechanical properties in sheet in the so-called hydrostatic bulging test. The chapter also shows equipment designed to obtain an effective stress strain curve by bulging a circular diaphragm with hydrostatic pressure. For punches that are not hemispherical, the strain distribution depends on the punch shape. Friction also affects the strain distribution and the greatest punch displacement. Normally, friction is confined to the punch comer radius. This prevents the tension across the face of the punch from increasing sufficiently to stretch the material over the face of the punch. The maximum depth that can be formed is therefore less with friction than without it. Stretching over a punch is often a pre-form operation, where material is redistributed to obtain favorable conditions for the final stage. In the second stage, the shape of the punch is defined by the part design; but for the preform punch, there is some flexibility in the shape and the tool designer needs to consider both curvature and friction in arriving at a suitable tool.

9 – Stretching circular shells

The purpose of this chapter is to indicate how the true stress-strain curve derived from a tensile test can be applied to other deformation processes that may occur in typical sheet-forming operations. A common feature of many sheet-forming processes is that compared with the stresses in the plane of the sheet (the membrane stresses), the stress perpendicular to the surface of the sheet is small. If this normal stress is assumed zero, a major simplification is possible. Such a process is called plane stress deformation and the theory of yielding for this process is described in the chapter. There are cases in which the through-thickness or normal stress cannot be neglected and the theory of yielding in a three-dimensional stress state is described in an Appendix. In the chapter, it is shown that for simple (monotonic, proportional) plane stress processes, it is possible to determine at any instant the principal membrane stresses required for deformation, provided that the current flow stress σ f is known and also that either the stress ratio α or strain ratio β are known. The theory given in the chapter applies only for an instantaneous state in which the strain increment is small and the flow stress constant.

2 – Sheet deformation processes

Bending along a straight line is the most common of all sheet-forming processes; it can be done in various ways, such as forming along the complete bend in a die, or by wiping, folding or flanging in special machines, or sliding the sheet over a radius in a die A very large amount of sheet is roll formed where it is bent progressively under shaped rolls Failure by splitting during a bending process is usually limited to high-strength, less ductile sheet, and a more common cause of unsatisfactory bending is the lack of dimensional control in terms of spring back and thinning If the line of bending is curved, the adjacent sheet is usually deformed in the process and the sheet is either stretched, which may lead to splitting, or compressed with the possibility of buckling There are special cases where sheet can be bent along curved lines without stretching or shrinking adjacent areas, but these require special geometric design In this chapter, simple cases of bending along straight lines are examined for the elastic, plastic, and fully plastic regimens

Z. Marciniak

Papers

Mechanics of sheet metal forming

8 – Cylindrical deep drawing

9 – Stretching circular shells

2 – Sheet deformation processes

6 – Bending of sheet