The following assumptions are made in using the slab method of analysis:
● The deforming material is isotropic and in- compressible.
● The elastic deformations of the deforming material and the tool are neglected.
● The inertial forces are small and are ne- glected.
● The frictional shear stress, s, is constant at the die/material interface and is defined as s ⳱ f ¯r ⳱ m ¯r/ 3.
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● The material flows according to the von Mises rule.
● The flow stress and the temperature are con- stant within the analyzed portion of the de- forming material.
The basic approach for the practical use of the slab method is as follows:
1. Estimate or assume a velocity or metal flow field.
2. For this velocity field, estimate the average strains, strain rates, and temperatures within each distinct time zone of the velocity field.
3. Thus, estimate an average value of the flow stress,r,¯ within each distinct zone of de- formation.
4. Knowingr¯ and friction, derive or apply the necessary equations for predicting the stress distribution and the forming load (in the slab method) or the forming load and the average forming pressure (in the upper- bound method).
9.2.1 Application of Slab
Method to Plane-Strain Upsetting
The Velocity Field. In this case, deformation is homogeneous and takes place in the x-z plane (Fig. 9.1). The velocity field, with the velocities in the x, y, and z directions, is defined as: vz ⳱ ⳮV z/h; v ⳱ V x/h; v ⳱ 0D x D y (Eq 9.1) where VDis the velocity of the top die.
The strain rates are: vz VD ˙ e ⳱z z ⳱ ⳮ (Eq 9.2a) h vx VD ˙ e ⳱x ⳱ ⳱ ⳮ˙ez (Eq 9.2b) x h vy ˙ e ⳱y y ⳱ 0 (Eq 9.2c)
Fig. 9.2 Equilibrium of forces in plane strain homogeneous upsetting
It can be shown easily that the shear strain rates arec ⳱ ˙c˙xz yz⳱ 0.
The strains are: h
e ⳱ lnz ;e ⳱ ⳮe ; e ⳱ 0x z y (Eq 9.3) ho
The effective strain rate is given by the equa- tion:
2 2 2 2
˙¯e ⳱
冪
(˙e Ⳮ ˙e Ⳮ ˙e )x y z3 2 2 ˙ e Ⳮ ˙ex z 2 2 ˙¯e ⳱
冪
2冢
冣
⳱ |˙e | ⳱x |˙e |z (Eq 9.4) 3 冪3 冪3The effective strain is: 2
¯
e ⳱ |e |z (Eq 9.5)
3 冪
The slab method of analysis assumes that the stresses in the metal flow direction and in the directions perpendicular to the metal flow direc- tion are principal stresses, i.e.:
r ⳱ r , r ⳱ r ; r ⳱ rz 1 x 3 y 2 (Eq 9.6) Plastic deformation/plastic flow starts when the stresses at a given point in the metal reach a certain level, as specified by a flow rule such as the Tresca or von Mises rule discussed in Chap- ter 5. Analysis of plastic deformation requires a certain relationship between the applied stresses and the velocity field (kinematics as described by velocity, strain (e) and strain rate(˙e)fields). Such a relation between the stresses (in principal axes) and strain rates is given as follows:
˙ e ⳱ k(r ⳮ r )1 1 m ˙ e ⳱ k(r ⳮ r )2 2 m ˙ e ⳱ k(r ⳮ r )3 3 m
wherek is a constant and rmis the hydrostatic
stress.
These equations are called the plasticity equa- tions. From these equations, for the plane strain case one obtains:
˙ e ⳱ ˙e ⳱ k(r ⳮ r ) ⳱ 02 y 2 m or r ⳱ r2 m (Eq 9.7) Per definition: r Ⳮ r Ⳮ r1 2 3 r ⳱m 3 or, with Eq 9.7: r Ⳮ r1 3 r ⳱ r ⳱m 2 2
For plane strain, i.e.,r2⳱ rm, the von Mises
rule gives:
2 2 2
3[(r ⳮ r ) Ⳮ (r ⳮ r ) ⳮ 0] ⳱ 2 ¯r1 m 3 m
(Eq 9.8) After simplification, the flow rule is:
2 ¯r
r ⳮ r ⳱ r ⳮ r ⳱1 3 z x
冷 冷
(Eq 9.9) 3冪
Estimation of Stress Distribution. In apply- ing slab analysis to plane strain upsetting, a slab of infinitesimal thickness is selected perpendic- ular to the direction of metal flow (Fig. 9.2). Assuming a depth of “1” or unit length, a force balance is made on this slab. Thus, a simple equation of static equilibrium is obtained [Thomsen et al., 1965] [Hoffman et al., 1953].
Fig. 9.3 Equilibrium of forces in axisymmetric homogeneous upsetting
Summation of forces in the X direction is zero or:
F ⳱ r h ⳮ (r Ⳮ dr )h ⳮ 2sdx ⳱ 0
兺
x x x xor
dr ⳱ ⳮ2sdx/hx
Thus, by integration one gets: 2s r ⳱ ⳮx xⳭ C
h
From the flow rule of plane strain, it follows that:
2s 2
r ⳱ ⳮz xⳭ C Ⳮ
冷 冷
r¯ (Eq 9.10)h 冪3
The constant C is determined from the boundary condition at x⳱ ᐉ/2, where rx⳱ 0, and, from
Eq 9.9: 2 r ⳱z
冷 冷
r¯ 3 冪 Thus: 2s ᐉ 2 r ⳱ ⳮz冢
ⳮ x ⳮ冣
r¯ (Eq 9.11) h 2 冪3Equation 9.11 illustrates that the vertical stress increases linearly from the edge (x ⳱ ᐉ/2) of Fig. 9.2 toward the center (x ⳱ 0). The value ofrzis negative, because z is considered
to be positive acting upward and the upsetting stress is acting downward. Integration of Eq 9.11 gives the upsetting load.
In Eq 9.11, the frictional shear stress, s, is equal to m ¯r/ 3.
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Thus, integration of Eq 9.11 over the entire width,ᐉ, of the strip of unit depth gives the upsetting load per unit depth:2 ¯r mᐉ L ⳱
冢
1Ⳮ冣
ᐉ4h 3
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9.2.2 Application of the Slab Analysis Method to Axisymmetric Upsetting
Figure 9.3 illustrates the notations used in the homogeneous axisymmetric upsetting. The anal- ysis procedure is similar to that used in plane strain upsetting.
Velocity Field. The volume constancy holds; i.e., the volume of the material moved in the z direction is equal to that moved in the radial di- rection, or:
2
pr V ⳱ 2prv h, or v ⳱ V r/2hD r r D
In the z direction, vzcan be considered to vary
linearly while satisfying the boundary condi- tions at z⳱ 0 and z ⳱ h. In the tangential di-
rection,H, there is no metal flow. Thus, the ve- locities are:
vr⳱ V r/2h; v ⳱ ⳮV z/h; v ⳱ 0D z D H (Eq 9.13) In order to obtain the strain rate in the tan- gential direction, it is necessary to consider the actual metal flow since vH⳱ 0 and cannot be
used for taking a partial derivative. Following Fig. 9.3, the increase in strain in theH direction, i.e., the length of the arc, is given by:
(rⳭ dr)dH ⳮ rdH dr
de ⳱H ⳱ (Eq 9.14)
rdH r
or the strain rate is
deH dr 1 vr VD
˙
e ⳱H ⳱ ⳱ ⳱ (Eq 9.14a)
dt dt r r 2h
The other strain rates are: vz VD ˙ e ⳱z z ⳱ ⳮ (Eq 9.14b) h vr VD ˙ e ⳱r r ⳱ 2h ⳱ ˙eH (Eq 9.14c) 1 vr v ˙ c ⳱rz
冢
Ⳮ冣
⳱ 0 (Eq 9.14d) 2 z r ˙ cHz⳱ ˙c ⳱ 0rH (Eq 9.14e)Thus, the effective strain rate is:
2 2 2 2
˙¯e ⳱
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(˙e Ⳮ ˙e Ⳮ ˙e ) ⳱ |˙e |H r z z (Eq 9.15) 3The strains can be obtained by integrating the strain rates with respect to time, i.e.:
t tV dt D e ⳱z
冮
e dt ⳱ ⳮ˙z冮
to to h or withⳮdh ⳱ ⳮVDdt: h dh h e ⳱z冮
ⳮ ⳱ ⳮln (Eq 9.16a) ho h hoSimilarly, the other strains can be obtained as:
1 h ez
e ⳱ e ⳱H r ln ⳱ ⳮ (Eq 9.16b)
2 ho 2
In analogy with Eq 9.15, the effective strain is: ¯
e ⳱ |e |z (Eq 9.17)
The flow rule for axisymmetric deformation is obtained by using a derivation similar to that used in plane strain deformation. Because
, the plasticity equations give: ˙
e ⳱ ˙er h
r ⳱ r or r ⳱ rr h 2 3
Thus, the von Mises flow rule for axisymmetric upsetting is:
r ⳮ r ⳱ | ¯r| or r ⳮ r ⳱ | ¯r|1 2 z r (Eq 9.18) Estimation of Stress Distribution. The equi- librium of forces in the r direction (Fig. 9.3) gives [Thomsen et al., 1965] [Hoffman et al., 1953]: F ⳱ r (dh)rh ⳮ (r Ⳮ dr )
兺
r r r r dh (rⳭ dr)hdh Ⳮ 2r sinh 2 hdrⳮ 2srdhdr ⳱ 0 (Eq 9.19) The angle dh is very small. Thus, with sin dh/2 ⳱ dh/2, and after canceling appropriate terms, Eq 9.19 reduces to:drr 2s
ⳮr ⳮr dr r Ⳮ r ⳮh h r ⳱ 0 (Eq 9.20)
Since in axisymmetric deformation, ˙e ⳱ ˙e ,r h
the plasticity equations give: drr 2s r ⳱ r , orr h Ⳮ ⳱ 0 (Eq 9.21) dr h Integration gives: 2s r ⳱ ⳮr r Ⳮ C h
The constant C is determined from the condition that at the free boundary, r⳱ R in Fig. 9.3, and the radial stressrr⳱ 0. Thus, integration of Eq
9.21 gives: 2s
r ⳱r (rⳮ R) (Eq 9.22)
With the flow rule, Eq 9.22 is transformed into:
2s
r ⳱z (rⳮ R) ⳮ ¯r (Eq 9.23)
h
Equation 9.23 illustrates that the stress increases linearly from the edge toward the center. The upsetting load can now be obtained by integrat- ing the stress distribution over the circular sur- face of the cylindrical upset:
R
L⳱
冮
r 2prdrz 0Considering thats ⳱m ¯r/ 3,
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integration gives: 2mR2
L⳱ ¯rpR 1 Ⳮ
冢
冣
(Eq 9.24)3h 3冪