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14. Slab Analysis of Bulk Forming Processes

ME 612 Metal Forming and Theory of Plasticity

- Assoc.Prof.Dr. Ahmet Zafer Senalp e-mail

azsenalp_at_gmail.com - Mechanical Engineering Department
- Gebze Technical University

14. Slab Analysis of Bulk Forming Processes

- This method entails a force balance on a slab of

metal of differential thickness. This produces a

differential equation where variations are

considered in one direction only. Using pertinent

boundary conditions, an integration of this

equation then provides a solution. The

assumptions involved are the following - 1. Friction does not influence the orientation of

the principal axes. In the Figures below we

assume that x,y,z are fixed principal axes within

the deformation zone. - 2. Plane sections remain plane, thus the

deformation is homogeneous in regard to the

determination of induced strain. - 3. The principal stresses do not vary on the yz

plane (see Figure below)

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

Figure 14.1. An example of sheet drawing showing

a slab. The axes X, Y, Z are assumed to be

principal stress axes.

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- The drawing stress is defined as follows
- Using incompressibility
- and plane strain conditions,
- results in the following
- We define the homogeneous strain as

(14.1)

(14.2)

(14.3)

(14.4)

(14.5)

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

Figure 14.2. Sheet Drawing. Free body diagrams to

calculate and

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- Taking in the left free body

diagram, - And since
- In general, ltlt 1 so
- indicating that is compressive.
- Using the flow rule for, we have

that or finally using equation

(14.7) and the definition of the deviatoric

stress, - In terms of principal stresses, since is

obviously tensile, then - Substitution in the von-Mises criterion, gives

the following relation between the stress

components to initiate and sustain plastic

deformation (plane strain)

(14.9)

(14.7)

(14.8)

(14.9)

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- or
- This equation is valid in any location inside the

deformation zone but we here assume a

nonhardening material (Y - constant). - Considering equilibrium of forces in the X

direction - or after neglecting higher order terms,

- and using dt/2 ds sina we finally have
- Let us define B as follows
- The equilibrium equation is now simplified as

follows

(14.10)

(14.11)

(14.12)

(14.13)

(14.14)

(14.15)

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- Substituting p from equation (14.10) in the

equation above gives - The solution is based on the following

assumptions. - An average constant value of µ describes the full

contact region. - The metal does not work harden, or a mean value

of yield stress strength adequately describes any

work-hardening effects in either case,Y is

treated as a constant. - The semi-die angle a is a constant.
- Direct integration using the conditions that

when and when - gives

(14.16)

(14.17)

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- or using homogeneous strain,
- olur.
- Note Consider (i.e. no

friction). Then using a Taylor series expansion

of the exponential term in equation (14.18) leads

to - which as gives
- .

(14.18)

(14.19)

(14.20)

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- which is the answer also provided by the ideal

work method!! From this it can be realized that

the slab analysis method simply extends the

information provided by the ideal-work method to

include frictional effects. - Example
- A sheet of metal having an initial thickness of

0.100 in. and width of 12 in. is to be drawn

through straight-sided dies having an included

angle of 300. If the average of the yield stress

is ksi and an average value for the

coefficient of friction is 0.08, calculate the

force needed to complete this operation for a

reduction of 10. - Solution
- From where a is the

semi-die angle,

14.1. Plane Strain Drawing

14. Slab Analysis of Bulk Forming Processes

- The drawing force , so

(14.21)

(14.22)

(14.23)

(14.24)

(14.25)

ksi

lbf

(14.26)

14.2. Wire or Rod Drawing

14. Slab Analysis of Bulk Forming Processes

- In terms of principal stresses,
- where z is the main axis and the directions 2 2

and 3 3 are the hoop and radial directions.

Substitution of the above equation in the

von-Mises criterion, gives the following relation

between the stress components to initiate and

sustain plastic deformation (axisymmetric

problems) - or
- For a wire or rod of circular cross section the

basic governing equation is - so that,
- Integrating gives

(14.27)

(14.28)

(14.29)

where

(14.30)

14.2. Wire or Rod Drawing

14. Slab Analysis of Bulk Forming Processes

- where

(14.31)

(14.32)

Figure 14.3. Slab analysis for rod drawing.

14.3. Plane Strain Extrusion

14. Slab Analysis of Bulk Forming Processes

- We here list only the equations that are

different from those of plane strain drawing

Figure 14.4. Plane strain and axisymmetric

extrusion.

14.3. Plane Strain Extrusion

14. Slab Analysis of Bulk Forming Processes

- Axisymmetric extrusion
- Notes
- The equations above are for a non-hardening

material. In the case of a hardening material,

one can use the above equations (as an

approximation) by taking Y to be the mean yield

stress over the range of strain induced by the

shape change. - It should be noted that these analyses become

unrealistic at high die angles and low

reductions. Assuming that P is a principal stress

is reasonable only if a is small and friction is

low. - All slab analysis calculations do not account for

redundant (non-homogeneous) deformation.

(14.33)

(14.34)