Multiaxial

Forming Limit Diagrams

Note the three limiting strain paths: equibiaxial, plane-strain and uniaxial. The forming limit is the point of fracture which depends on the state of stress, with the plane strain having the lowest strain to fracture

Zinc

On the left is a picture of cylindrical single crystal of zinc which is pulled uniaxially.

(i) Plastic deformation is accommodated by sliding on crystal planes, without changing the crystal structure,

(ii) What is not immediately evident is the the direction of slip is also specific to a specific lattice translation vector,

(iii) The distance between the slip planes is much greater than the atomic spacing of the planes in teh crystal; the latter is sub-nanometer where as the "slab thickness" is several micrometers, (iv) the slip distance can be translated into uniaxial elongation from geometry.



An interesting point about this micrograph is that it is seen only with single crystals. If a polycrystal of zinc is deformed it falls to pieces as if it is brittle. As seen later this happens because of paucity of slip systems in the crystal structure of zinc

Critical Resolved Shear Stress

CRSS recognizes that slip occurs on a specific plane and direction that may have any orientation with respect to the uniaxial pulling direction. The material parameter is the yield stress for a specific slip system, however. The analysis below shows how the measured uniaxial yield stress can be related to the "crystallographic" yield stress. It is related to the orientation of the slip plane and the slip direction with respect to the uniaxial direction

Ideal Shear Yield Stress in a Perfect Crystal

The following pages give the derivation for the ideal shear yield stress of a perfect crystal assuming a sinuosidal force-displacement function. Note that the ideal yield stress varies with the crystal plane and the slip direction

Slip Systems in Cubic, BCC and FCC Crystalss

Measurments of the Yield Stress

The ideal shear yield stress of perfect crystals amounts to about 10% of the shear modulus. As shown on the left the actual yield stress of metals can be several orders of magnitude lower.

Tensile stress of whiskers: needle like single crystals depends on the diameter of the whiskers. The yield stress keeps on rising as the whisher diameter decreases, approaching the ideal yield stress because defects are less probable as the specimen size decreases

Description and Properties of Dislocations

Dislocations have the following features:
(i) They are "line" defects that can explain the greatly lowered yield stress in real crystal.
(ii) Dislocations are characterized by two vectors, one is the line that describes the dislocation, and the other is the slip vector, also called the Burgers Vector that is associated with the dislocation.
(iii) The slip vector may be parallel to or perpendicular to the line of the dislocation.
(iv) Dislocation can assume a curved character although the line must reside in one slip plane for the dislocation to slide within the slip plane. (v) Dislocations can slip at very low applied stress.
(vi) There is a characteristic spacing among the dislocations.
(vii) The macroscopic plastic shear strain is related to the density of the dislocations, their Burgers Vector, and the slip distance for each of these dislocations

Dislocation Mechanics

Dislocation Stress Fields, Reactions and Multiplication

Considere' Condition: onset of strain localization