Cambridge University Engineering Department

ENGINEERING TRIPOS PART IIA ENGINEERING TRIPOS PART IIA MANUFACTURING ENGINEERING TRIPOS PART IIA

Module 3C1: Materials processing and designModule 3P1: Materials into products

WROUGHT ALLOY PROCESSING

1. Introduction to wrought alloy deformation processes1.1 Overview of deformation processes1.2 Deformation mechanisms: hot vs. cold working

2. Fundamentals of plasticity – (mostly) revision2.1 Stress states in 3D2.2 Yield criteria2.3 Plastic strain and flow rules2.4 Plane strain conditions

3 Equilibrium analysis3. Equilibrium analysis3.1 Forging3.2 Sheet and wire drawing3.3 Rolling – forging analogy

4. Upper bound analysis4.1 Forgingg g4.2 Extrusion4.3 Machining

5. Temperature rise in deformation processing

6. Microstructure evolution in wrought alloy processing6 1 i l h i i f h ll6.1 Microstructural characteristics of wrought alloys6.2 Case Study: Extrusion of heat-treatable Al alloys

References

Kalpakjian S., Manufacturing Processes for Engineering Materials, Addison Wesley, CUED Library JN49y

Edwards S.L. and Endean M., Manufacturing with Materials, Butterworths, CUED Library JA146

Cambridge Engineering Selector (CES EduPack 2011 or 2012)

Web-based resource on Aluminium Technology: AluMATTERhtt // l i i tt k

1

H.R. ShercliffOctober 2012

http://www.aluminium.matter.org.uk

1. Introduction to wrought alloy deformation processes

1 1 O i f f i1.1 Overview of deformation processes

Deformation processing (or forming) is the shaping of material in the solid state:

Machining is also a plastic deformation process, used for refining shape or finish, and adding features (holes, threads etc.).

Each process class is used for a fairly specific geometric shaping activity, so there is little direct competition between them. But other processes compete with forming, e.g.

forging vs casting vs powder processing

Why carry out deformation processing?1. Geometry: forming long, thin-walled shapes (i.e. high aspect ratio, difficult to cast). 2. Low waste: forming processes mostly “near net shape”.3 T l d f fi i h ll d ( d b t d b hi i )

- forging vs. casting vs. powder processing- extrusion vs. welded assembly of plate

3. Tolerance and surface finish: usually good (and can be corrected by machining)4. Microstructure: cast microstructures usually coarse – need to be refined by

deformation (and heat treatment) to enhance properties.5. Energy and cost efficient: temperatures below melting.

Some disadvantages:Some disadvantages:1. High forces and complex control systems are required: can be high capital cost,

with expensive high strength steel tooling.2. Multiple stages (including machining) often needed due to physical limits on

achievable shape changes and complexity.3. Metals work-harden with cold deformation and often require intermediate annealing

to enable further deformation.

2

to enable further deformation.

1.2 Deformation mechanisms: hot vs. cold working

Key aspects of understanding how processes work (e.g. to control and model them):

- deformation geometry (plane strain, axisymmetric, 3D)

- tool/workpiece interactions (friction, heat transfer)- temperature history (governs deformation mechanism)

Steady-state processes Non-steady-state processes(fixed deformation pattern through

which material flows)(component geometry and deformation

pattern continuously changes)

Rolling hot or cold Forging usually hot

Extrusion usually hot Sheet metal forming cold

Wire drawing coldWire drawing cold

Machining cold

“Hot” and “cold” refer to whether the material is pre-heated before forming – but all deformation generates heat, so even a cold-worked part may undergo some heating. Th d i f i d i h d f i h iThe average temperature during forming determines the deformation mechanism.

Cold working Hot working

plastic yielding:T < 0.3 Tm, typical = 1–105 s-1

high strain–rate creep:T > 0.5 Tm, typical = 1–103 s-1

T , : little influence on yield response T , : strong influence on yield response

deformed, elongated grains; anisotropic mechanical properties

recrystallised, equiaxed grains; isotropic mechanical properties

modest forming in tension viable (work hardening suppresses necking)

must be worked in compression (dynamic softening: necking in tension)

( g pp g) ( y g g )

good surface finish and tolerance poor surface finish (oxidation) and tolerance (differential thermal contraction)

How hot is hot?

Material Hot working temperatures (0.5 – 0.95 Tm)

Steels 800 – 1300 oC (i.e. austenite in C steels)

Al alloys 300 – 500 oC

3

Elastic-plastic stress-strain curves for cold deformation in tension (Materials Databook):

Constitutive behaviour (stress-strain response)

- cold deformation leads to work hardening

- elastic strains small ( 0.1%), and may be neglected for most forming processes

NB. forming processes (hot or cold) are mostly compressive (to avoid the necking instability in tension), allowing much larger plastic strains, typically 10s to 100s of %.

For processing at high temperatures (hot working), the stress-strain response is typically:

), g g p , yp y

A steady-state (constant) flow stress is a result of a balance between dislocation accumulation (work-hardening) and dislocation annihilation (by dynamic recovery). This is typical of

aluminium alloys.

Other alloys (steels) undergo dynamic recrystallisation – new grains forming, growing, and work hardening in a continuous cycle (again reducing the flow stress).

Dynamic recovery and recrystallisation depend on both the strain-rate and the temperature (i.e. diffusion-controlled thermally activated processes), see figure.

4

Modelling of forming processes has many potential applications and benefits:- co-design of components and the forming equipment used to make them

(t i fid th t th ill k tti t d t )

Plastic analysis of forming processes

(to give confidence that the process will work, cutting costs and scrap rates)- offline evaluation of sensitivity to fluctuations in process parameters, in order to determine

optimum set-points for these parameters (e.g. metal temperature, process speeds, friction conditions)

- prediction of temperature and deformation histories throughout a component, as input to understanding or modelling the distribution of microstructure and properties.

Most modelling of deformation processes uses advanced numerical methods (finite element methods, FEM, or computational fluid dynamics, CFD), combining metal flow and heat flow. This enables detailed prediction of the behaviour of a given design processed in a particular way (provided the models, input data and implementation are robust!).

A l i l h d i i lifi i f h bl b i i k ll f

The simple analytical calculations here predict forces, energy input, and temperature riseduring deformation processing, using two approaches:

Analytical methods require simplification of the problem, but give a quick overall sense of how the processes work, and the influence of key process variables. They also provide simple checks on sophisticated numerical analyses.

g p g, g pp

• Equilibrium analysis: find a stress field which satisfies equilibrium and produces yielding (hence flow) at every point.

• Upper bound analysis: find a displacement field which satisfies compatibility and allows the deformation to occur. Equate internal and external work done.

Numerical methods can handle the complex elastic-plastic stress-strain behaviour (such as strain-rate and temperature-dependent yielding).

For analytical methods, we approximate the uniaxial stress-strain curve as rigid

Elastic limit, uniaxial yield stress, Yg

perfectly-plastic (i.e. neglect elastic strain, and assume a constant yield stress).

infinite Young’s modulus, EStr

ess

plastic flow

Deformation processing is inherently three-dimensional, so the analysis also requires:

- a yield criterion applicable to any stress state, determining when plastic flow occurs

- a flow rule relating the resulting plastic strains (or strain-rates) to the stress state.

Strain,

5

2. Fundamentals of plasticity (mostly revision of IB Structures)

2.1 Stress states in 3 dimensions

zz general stressstate

in terms of principal stresses

yzyyyx

xzxyxx

2

1

00

00

00

σ1

3

xyyy

xx

yx zzzyzx 300

σ2

For any general stress state we can find a set of i i l Th t t f th

(yy, xy)

principal axes. The stress tensor for these axes contains no off-diagonal (shear) terms – only three principal stresses along the three axes.

Mohr’s circle allows rotation of axes in two dimensions about one principal axis

21

2θ

(xx, -yx)

2.2 Yield criteria

The yield criterion defines the limit of elasticity under any possible combination of stresses.

The mean or “hydrostatic” stress is defined as .

The hydrostatic stress has no effect on yielding in metals (although it does affect yielding in polymers, and also affects fracture). We can separate any stress state into its hydrostatic and deviatoric components:

)( 32131 m

p

m

m

m

m

m

m

3

2

1

3

2

1

00

00

00

00

00

00

00

00

00

The yield criterion becomes a condition about the deviatoric stress tensor – i.e. yield occurs when some function of the deviatoric stress tensor reaches a critical value.

6

Tresca yield criterionYi ld h th i h t h iti l l T ki Y th i ldYield occurs when the maximum shear stress reaches a critical value. Taking Y as the yield stress in uniaxial tension, and using Mohr’s circle, this gives:

Usually define the principal stresses so that . Hence: 321

Y 133221 ,,max

y p p 321

31 Y

Von Mises yield criterion According to the von Mises criterion, yield occurs when the elastic shear strain energy reaches a critical value, which leads to :,

2213

232

221 2 Y

Both criteria can also be written in terms of the Mohr’s circle for pure shearBoth criteria can also be written in terms of the yield stress in pure shear, k.

The principal stresses are then

1 = + k; 2 = 0 ; 3 = - k

Mohr s circle for pure shear

k

13

k

k

so the two yield criteria can be written as:

Tresca: 1 – 3 = Y = 2k

k

von Mises: (1 – 2)2 + (2 – 3)2 + (1 – 3)2 = 2Y2 = 6k2

NB. the relationship between Y and k changes: for Tresca, Y = 2k ; for von Mises Y = √3k

i ibl i ll hi h i i i b b i d k hIt is possible to test experimentally which criterion is better, by measuring Y and k on the same sample of material (and under combined tension and shear). The differences between the two criteria are relatively small (at most 15%).

7

2.3 Plastic strain and flow rule

Plastic deformation conserves volume. Thus, in terms of principal strain increments:

0 321 ddd

Once the yield criterion is satisfied we need a flow rule to relate the plastic strain to the stress. In an isotropic material the principal axes of stress and strain-rate coincide.

The Levy-Mises flow rule states that the plastic strain increment in each principal direction is proportional to the deviatoric stress component in that direction i.e.

mmmddd 321321 ,,,,

dddThis can be written as:

mmm

ddd

3

3

2

2

1

1

Written in terms of the three principal stresses only, it becomes: )()()( 212

13

3

3121

2

2

3221

1

1

ddd

2.4 Plane strain conditions

In plane strain, one dimension of the deforming geometry is much larger than the others and there is no strain in that direction (giving 2D in plane deformation)there is no strain in that direction (giving 2D, in-plane deformation).

Plane strain is a common simplifying assumption in forming, e.g. for rolling, slab extrusion or drawing, some forging problems, and machining.

If the principal strain in the 2 direction is zero (i.e. 2 = 0), then from the Levy-Mises equation:

finiteandzeronon2 d

finite and zero-non)( 312

12

Thus: 2 = ½ (1 + 3).

i.e. 2 will be the intermediate principal stress and 1 > 2 > 3

The maximum shear stress will be ½ (1-3) on planes at 45° to the 1 and 3 axesThe maximum shear stress will be ½ (1 3) on planes at 45 to the 1 and 3 axes

By the Tresca yield criterion: 1 – 3 = Y = 2k

By the von Mises yield criterion: 1 – 3 = 2Y/√3 = 2k

So for plane strain, the two yield criteria are the same, in terms of the pure shear yield stress k.

8

Using the uniaxial yield stress Y, they differ by about 15%. The higher value governing yield by the von Mises criterion, 2Y/√3 , is sometimes called the “plane strain yield stress”.

3 Equilibrium analysis

3.1 Forgingg g

Open die forging

Impression / Closed die forging

Consider open die forging of a long rectangular slab using wide dies. The slab and dies are long in the z direction, with plastic flow restricted to the xy plane – i.e. plane strain conditions.

y

Analysis of plane strain forging (revision of IB Materials)

Fp(x)

xz

x

τ(x)

( )

dx2h

Take unit depth into the diagram

Slab width 2w, height 2h

For convenience, take +ve when τ(x)

w

,compressive for this problem.

Assume Coulomb friction: = p

Purpose of calculation: find forging load F. Approach: find pressure distribution at die face p(x) and integrateApproach: find pressure distribution at die face p(x) and integrate.

dx

τ(x)p(x)The stress state is non-uniform – consider

the vertical mid-plane: this must be loaded in compression, to balance the inward friction forces on the top and

τ(x)( )

σx (σx + dσx)2hbottom of each half of the billet.

Hence we must consider the equilibrium of an element of width dx, height 2h

9

τ(x)p(x)

Equilibrium in the x-direction:

hdx

d

dxhhd

x

xxx

0.22.2.

Principal directions:

At the element centre, there is no shear (due to symmetry): x and y are principal stresses.

At the element surface, there is shear stress due to friction – but consider the Mohr’s circle for a moderate value of :

hdx

Hence to a good approximation, x and y (= p) are also principal stresses at the surface.

Recall that for plane strain:

0dz is the intermediate principal stress

i.e. principal stresses: p > z > x

0zzd

0dd p

Y xYield criteria:

Fi t id T 0dd

xx

pYp x

x

Friction law: Coulomb friction:

First consider Tresca:

Hence:hx

p

d

d

p

Hence:

Separate variables and integrate:

h

p

x

p

d

d

Cxh

μ(p)dx

hp

dp

ln

Boundary conditions:

At x = w, x = 0. Combining this with the yield criterion p – x = Y gives at x = w, p = Y.

Hence: and thus:Cwh

μY )(ln )0(exp

xxw

h

μ

Y

p

When x < 0 the direction of friction is reversed, and the solution becomes:

0exp

xxw

h

μ

Y

p

10

NB: if we had used the von Mises criterion, , the expressions for p/Ywould be increased by the factor of 2/√3.

3/2 Yp x

w = 4h

5 0

w = 2h

5 0

‘friction hill’

1 0

2.0

3.0

4.0

5.0

p/Y

1 0

2.0

3.0

4.0

5.0

p/Y

μ = 0.4

0.0

1.0

-1 -0.5 0 0.5 1

x/w

0.0

1.0

-1 -0.5 0 0.5 1

x/w

μ = 0.1

To calculate the total forging load F, integrate p(x) over the die surface (depth D into page):

1exp

2 )(

h

wDhYdxxpDF

w

w( 2/√3 using von Mises)

Note that the target result – the load required to yield the slab – depends on a combination of:- material parameters: yield stress, Y- process operating conditions: friction between die and workpiece, - design parameters: the geometry of the slab – depth D, height h, width w

Note in particular the sensitivity to the aspect ratio, w/h:- the high value of p/Y sets a limit on how thin a component can be forged- the thin flash formed at the edge in impression die forging accounts for a significant share of

the total load

Plane strain compression without friction

In the limit 0 (frictionless dies), the results become: p = Y, and F = 2DwY (using Tresca’s criterion). Check this for yourself!

Recall that by the Tresca criterion, p = Y = 2k , where k is the shear yield stress.

Furthermore, by the von Mises criterion, p = (2/√3) Y = (2/√3) k/√3 = 2k .

So the pressure on the dies in plane strain compression with frictionless dies is 2k (whichever criterion is used). This geometry can therefore be used in a compression test to measure 2kfor the material directly.

The conversion to uniaxial yield stress Y depends on whether the material response is closer

11

The conversion to uniaxial yield stress Y depends on whether the material response is closer to Tresca or von Mises.

Summary of the equilibrium method for finding forming pressures/loads:

1. Assume a set of (approximate) principal directions.

2. Assume a friction law. The usual choices are : Frictionless (τ = 0), Coulomb friction (τ =μp), or sticking friction (τ = k, where k is the shear yield stress.)

3. If the stress state is not uniform, consider the equilibrium of an element to get adifferential equation relating stresses in the direction of variation.

4 Assume a flow rule and yield criterion (N B the difference between von Mises and4. Assume a flow rule and yield criterion. (N.B. the difference between von Mises and Tresca amounts to only 15% difference in the predicted stresses for plane strain).

5. Use the flow rule and yield criterion to relate the principal stresses.

6. Decide which principal stress is required for the problem in hand – e.g. for the verticalforging load we needed p(x). Thus we eliminated the other unknown stress x to i diff ti l ti i th i d tgive a differential equation in the required stress, p.

7. Solve for the variation of this stress in the x direction by integrating subject to appropriate boundary conditions.

8. For loads on the tooling, integrate the pressure over the tool area.

Forging of a disc (not on syllabus)

Exactly the same principles apply for the analysis of forging of a disc. The friction acts radially inward giving an axisymmetric friction hillinward, giving an axisymmetric friction hill.

In this case, all three strains are non-zero (i.e. in the r, z and directions). A further relationship between the strain components (i.e. volume conservation) is needed to find and eliminate the second unknown principal stress (in the hoop

The resulting pressure distribution p(r) is again integrated to find the forging load:

direction,

22

21exp2 hRh RYYF

Note the similarities to the plane strain case – the forging load depends on Y, , h and R, and is particularly sensitive to the thickness to diameter ratio, h/2R.

22

21exp2h

RYYF

12

3.2 Sheet and wire drawingTo reduce the thickness of a long product requires a continuous process, in which the material is forced through shaped tooling . In sheet drawing, a flat strip is pulled through a profiled die with the die exit determining the final dimensions; the axisymmetric equivalent is wire drawing.

C id th l t i d i f id h t b t f i ti l di

12

3w

ho

Consider the plane strain drawing of a wide sheet between frictionless dies:

hi σdraw

Assume zero friction.

Take compressive stress +ve.

Depth D into the page is large, so plane strain conditions apply, with d2=0.

zoom in on detail….

p

d

sin2

d hp

d hp

111 d

2

d h

2

d h

tan2p

2

dhp

3 tan2

Equilibrium: Resolve horizontally.

0ddd

02

d

22

dd

11

111

hphh

hp

hhh

Resolve vertically. ph

ph

33 tan2

d

tan2

d

Yield criterion: no flow in the 2 direction so 2 is intermediate; the Tresca criterion gives:

YpY 113 hence

NB: In this case we want the stress in the x-direction (at exit), rather than the distribution of pwith x, so we don’t switch to a differential in p, but eliminate p leaving a differential in 1.

Combining the yield criterion with horizontal equilibrium: 0dd 1 hYh

Integrate and use boundary conditions at entry ( = 0) and exit ( = - d ):

13

Integrate and use boundary conditions at entry ( 0) and exit ( draw):

o

idraw

h

h h

hY

h

hY

i

o

lnd

d0

-1

draw

Notes:

1. This analysis used the Tresca criterion – using von Mises instead, replace Y with 2Y/3(as plane strain)

2. Note that σdraw may not exceed the yield stress Y (or the material fails in tension at theexit). This gives a maximum draw ratio:

43%) :Mises(for von %37 i.e. 718.21lnlnmax

i

o

o

i

o

i

o

i

h

he

h

h

h

hY

h

hY

max

3. A back tension σback may also be applied at the inlet. This has no effect on the initial analysis but changes one of the boundary conditions: at the inlet, h=hi , σ1 = -σback . Hence:

o

ibackdraw h

hY ln

So the draw stress is increased by the back tension, reducing the maximum draw ratio. But increased tension along the strip reduces the pressure needed to reach yield – hence back tension reduces wear of the die.

oh

4. The same analysis can be applied to plane strain extrusion, simply by another change in boundary conditions: at the inlet, σ = + σext (compression), and at the outlet σ = 0.

The inlet material must then be enclosed, but the compressive stress is not limited by theyield stress – so greater reductions in section are possible.

5. The analysis can readily be extended to include the effect of friction on the die (giving additional terms in the equilibrium equations).

Wire drawing

Wires are drawn in tension through a sequence of well-lubricated conical dies. Typical reduction in area per pass = 10–40%. Work hardening is neededarea per pass 10 40%. Work hardening is needed to avoid tensile failure at the die outlet, but the material may then require intermediate annealing.

Wire drawing may be modelled using an equivalent axisymmetric analysis to sheet drawing.The resulting expression for the draw stress, for the case of Coulomb friction on the die wall, is:The resulting expression for the draw stress, for the case of Coulomb friction on the die wall, is:

cot

1

2

A

A1

cot

11Ydraw

Note that the observations above on the maximum draw ratio and the effect of back tension apply equally to wire drawing and a change in boundary conditions to an inlet compression

14

apply equally to wire drawing, and a change in boundary conditions to an inlet compression switches the analysis to extrusion of a solid cylinder.

3.3 Rolling

Rolling is a steady-state, continuous process for forming long prismatic shapes.

Cast Ingot

Flat Rolling (sheet, strip, plate) Section Rolling (I-beams etc)

Cast Ingot

"Bloom“ or "Slab"

Rolling mill designs:

Reversing mill: material passes backwards and forwards through the same mill stand, which is incrementally closed before each pass. Used for “breakdown” rolling of ingot/thick slab, and f i llifor section rolling.

Tandem mill: several mill stands (typically four) in connected series. Used for strip/sheet/foil, i.e. stock material that is thin enough to be coiled after rolling. The strip accelerates considerably as its thickness is reduced (due to conservation of volume).

Analysis of rolling: by analogy with forgingAnalysis of rolling: by analogy with forging

Rolling of wide strip or plate can be thought of as an adaptation of plane strain drawing in which the tapered gap between stationary dies is replaced by the gap between rotating rolls. In this case some friction between rolls and strip is essential, since it is friction that draws the material into the roll bite (rather than a drawing stress at outlet, or compression at inlet).

2wll di R

hi

vi

ho

vo

ωroll radius R

F

τ F

Note that the friction direction reverses in the roll bite at the neutral plane, i.e. the point where

15

the local speed of the strip = the peripheral roll speed, v = R .

Note also that conservation of volume means that hivi = hovo .

F

22 oi hh

h

The contact patch, width 2w, is actually relatively small compared to the radius R; and the reduction per pass is modest (the figure above is exaggerated).

2w

2The deforming volume may be approximated to a rectangular block, width 2w, and height 2hgiven by the average of the inlet and outlet thicknesses.

Hence the rolling pressure distribution will be a friction hill, as in forging, and we can estimate the rolling load directly using the result for plane strain forging (assuming the Tresca criterion):

12 wDhY

F

The roll bite is therefore directly analogous to the geometry used for analysis of plane strain forging, assuming the neutral plane falls on the plane of symmetry.

1exp

hF

Provided the friction coefficient is reasonably low, we can assume w/h << 1. This allows the use of the approximation exp(w/h) 1 + w/h+ (w/h)2/2 , giving:

22 1DhY

It is more convenient to express the rolling force in terms of the roll radius R and heights hi andho. Consider the contact geometry:

2

22 1

2

DhY w wF

h h

oi

oi

Rhh

Rhh

Rw

giving 2

Typically

22 2

22

2w

R

2oi hh

oi hhRw 2

Substituting for w and h gives:

oi hhR

hhRYDF)(

)(

oi

oioi hh

hhRYDF)(

)(

As before, this result combines parameters of the material, process and the geometry of the strip.

Estimation of rolling torque and power

16

The roll closing force F acts at an offset distance w from the roll centre. The rolling torque is thus approximately T = F . w, and the power requirement (for each roll) is then P = T . for a rotational speed . This analysis is crude but gives the correct trends.

Controlling load and thickness

To reduce rolling loads:- reduce yield stress Y (i.e. hot rolling)

reduce friction (but note that some friction is- reduce friction (but note that some friction is required to draw in the strip)

- make small reductions per pass hi - ho (and tandem roll) - reduce roll radius R

Note that small radius rolls will bend, so larger backing rollsare used to reinforce the work rolls in a cluster mill designare used to reinforce the work rolls in a cluster mill design. Hydraulic jacks apply loads, continuously adjusted, to the roll stack. This ensures that a uniform material thickness (or gauge) is produced, both across and along the strip.

Cluster mill

Effects of tension at the inlet (back tension) and outlet (front tension)

Front and/or back tension are used in rolling, and have the effect of:- reducing the pressure needed to cause yielding (as in sheet/wire drawing)- reducing the magnitude of the friction hill (and shifting the neutral plane)- reducing the rolling load, torque and power

The analysis with forward/back tension follows the forging analogy, but with a change in boundary conditions at entry and exit.

Notes:

b f

1. If a sufficiently large back tension is applied, the neutral point can be shifted to the outlet, and the rolls begin to slip. This can be used experimentally to estimate the friction coefficient .2. In multi-stand tandem rolling, the forward tension on one stand = the back tension on the next. Why must this tension be carefully controlled?

17

4 Upper bound analysis

Upper bound theorem: Propose any mechanism of plastic collapse of a body and estimate the load required by equating the internal rate of energy dissipation to the rate at which the external l d d k h i d l d ill h b b l h l (i i illloads do work; the estimated load will then be above or equal to the correct value (i.e. it will represent an upper bound).

Any compatible mechanism of deformation may be chosen, but we can consider limiting cases in which all the deformation occurs in narrow shear bands, with the solid divided into rigid blocks sliding over each other, and the shear yield stress k acting at their interfaces.

The upper bound method with sliding blocks is highly effective for three reasons:(a) we do not have to satisfy equilibrium equations.(b) the upper bound loads are often surprisingly close to the correct collapse loads.(c) the load estimate is often insensitive to the exact choice of mechanism.

The method applies to 2-dimensional plane-strain problems only.

4.1 Forging

FvDepth D into page

Consider again plane strain forging of a long square block between frictionless dies:

A

BC D

E

A

h

B

C DE

Depth D into page

h

O (fixed)O (fixed)

F

Equate external rate of work with internal rate of energy dissipation:

External power = F . v

Internal power =

interfaceat velocitysliding relative stress yieldshear area Interface kinterfaces all

18

To find the relative sliding velocity of the blocks, construct a velocity diagram (hodograph):

o,e

Interface length velocity internal powerbc h / 2 v / 2 khDv / 2bd h / 2 v / 2 khDv / 2

vcd

ce h / 2 v / 2 khDv / 2ed h / 2 v / 2 khDv / 2

Total 2khDv

:power internal and external Equate

Reminder: rules for constructing a velocity diagram (hodograph)

1 Label all regions of the model which move relative to each other (convention: upper case)

a,bDhkFvDhkvF 22.

So pressure on die p = F/hD = 2k

1. Label all regions of the model which move relative to each other (convention: upper case)

2. Define an origin, corresponding to a stationary component of the system.

3. Draw the reference velocity vector of the unknown force.

4. Draw construction lines for the velocities of the moving blocks, either relative to the stationary components, or relative to each other. [Recall that the relative velocity between two sliding blocks must act parallel to the interface between the blocks Why is this? ]sliding blocks must act parallel to the interface between the blocks. Why is this? ]

5. The intersections of these construction lines give the velocity vectors of the labelled regions of the model (convention: lower case)

6. The relative velocities between sliding blocks can be scaled to the reference velocity, by measurement from a scale drawing, or trigonometry.

o

It is efficient to use the symmetry of the problem to reduce the number of shear planes to be considered, e.g. in the forging problem, take two mirror planes to split the problem vertically and horizontally – and consider the horizontal mirror plane to be stationary.

Fv/2

oc

v/2

A

B

C D

E

Ofixed

h √2 (v/2)

External power = 2 F . v/2 = F . v

Internal power = 4 (k D) BC Vb = 4 (k D) √2 (h/2) √2 (v/2)

a, b

O

Fv/2

19

Internal power 4 (k D). BC . Vbc 4 (k D) . √2 (h/2) . √2 (v/2)

= 2 k D h v

Hence F = 2 k h D as before.

Effect of aspect ratio on forging pressure

As we change the ratio of width b to height h in the forging problem, we can construct many different feasible deformation patterns. From the upper bound theorem, any kinematically correct pattern will constitute an upper bound – hence the pattern giving the lowest pressure will b l hbe closest to the correct answer.

The following four patterns are all kinematically admissible but each will be favoured for different ranges of h / b. Values of pressure p given are for the case of zero friction on the dies.

A: h >> b B: h > b

b

h/2

B: h > b

C: h ≈ b D: h << b

83.22

k

p

b

h

h

b

k

p

18

7

2

3

2

hbp 1 1p

bhk 22 2k

3

3.5

4

1.5

2

2.5

3

Pre

ssu

re p

/2k

A

B

C

20

0.5

1

0 2 4 6 8 10h / b

D

4.2 Extrusion

Extrusion uses compressive loading to force a billet (usually hot) through a die ( y ) gto make a shaped, prismatic section. With soft metals (e.g. hot aluminium) very large reductions in area can be made in a single step.

Die design

For solid cylinders, square or angled dies may be used. With a square die, a dead metal zone forms, leading to intense shearing (and heating).

The design of dies for complex hollow sections is considered in a later case study.

For upper bound analysis, we consider plane strain extrusion of a flat strip.

Upper bound analysis of plane strain extrusionConsider extrusion of a strip of thickness 2h from a billet of thickness 4h through a square die. Assume sticking friction on the die wall, and consider half of the problem (due to symmetry).The width of the strip (out of the page) is D. This geometry is referred to as direct extrusion.

External power = F.v/2

Ram

Die O

A B

CF/2 h

2vvD

l h

h

centre line

21

Velocity diagram:

InternalInterface length /h velocity /v power /(hDkv)

o,c v dvag y p ( )

OA l/h (oa) 1 l/hAB 2 (ab) 1 2BC 2 (bc) 2 2BD 2 (bd) 2 2

Total l/h + 6b

Total l/h + 6

hlkDFh

lvkDhvF 6262/.

Equate external and internal power:

Consider an alternative die design: indirect extrusion

Die O

fi d di

O, moving die

Ram

2vv

A B

C

D

F/2

h

fixed die

l h

The analysis is identical, except that O now moves with A so no shear work is done on interface oa.

hkDF 62Hence:

Machining processes use a hard tool to selectively remove a softer material. Most components undergo some machining. Numerically controlled machines give high reproducibility and

4.3 Machining

g g y g g p yaccuracy.

There is a wide range of machining processes: turning, shaping, milling, drilling, tapping, grinding. The underlying mechanism is largely the same: plastic deformation of a thin surface layer or groove of material.

Large shear strains are imposed on the chip and strain-rates are high (103 s-1) – this can cause

22

g p p g ( )temperature rises of the order of 1000C in steels.

Machining – upper bound analysis

Consider the idealised machining geometry below – orthogonal machining – with in-planedeformation, and a depth into the page which is larger then the chip thickness (i.e. plane strain).First, neglect friction on the tool face – giving a single primary shear zone.

α

C

T

Velocity diagram:

)(2

)2

(

φ

v

d

F

Rake angle

c

αvcw

FW

vt wφπ/2-α

:rulesin theusing Find vcw

:power internal and external Equate

)cos(

cos

)2/sin)(2/sin

vv

vvcw

cw

: respect to with maximum a is )cos(sin when iswhich of valueminimum thebe illsolution w optimal The

)cos(sin

cos i.e.

)cos(

cos

sin..

sin.

fF/D

dk

D

Fk

vD

dkvD

dvF cw

0)2cos()cos()sin(sin)cos(cos f

cos2/4)./2( :solution Principal

dkF

sin1

cos.2:givesfor expression theinto ngSubstituti

dk

D

FF

In practical machining, there is a secondary shear zone associated with friction on the rake face of tool (neglected above). This can be included by adding a term for the power dissipation on the tool, e.g. assuming sticking friction over the length of the tool face, with a relative velocity given by the chip exit velocity

23

given by the chip exit velocity.

5. Temperature rise in deformation processing

Effect of temperature rise during deformation processing

All of the plastic work is dissipated as heat. The workpiece temperature depends on:th l ti f th t l d t li- thermal properties of the metal and tooling;

- component geometry;

- heat transfer conditions (to the tooling, coolant or atmosphere).

Influences of temperature history:

• variations in T, and lead to variations in microstructure, and properties;

Estimate of temperature rise

• severe surface deformation can cause local melting and a poor surface finish (e.g. extrusion) or tool damage (e.g. machining);

• temperature rise may be essential for a combined deformation and heat treatment (e.g. extrusion of heat-treatable aluminium alloys).

Estimate of temperature rise

Many of the analyses led to estimates for the power input, . For example:

- rolling: power (per roll) = torque angular velocity

- drawing, extrusion, machining: external power in upper bound analysis

From conservation of energy, assuming adiabatic heating and a uniform temperature rise:

Q

CvA

QT

capacity)heat c(volumetri second)per deformed (volume

inputpower

How accurate is this approach?

1. The assumption of adiabatic heating will lead to an over-estimate, since there will be some heat loss to the tooling.

2. If all of the heating occurs in primary shear zones, through which all the material passes, the temperature rise

e.g. workpiece and tool temperatures in machiningp , p

will be reasonably uniform. But if significant heating occurs in secondary shear zones at the surface (i.e. due to friction on the tooling), a temperature gradient will be set up, and the peak temperature rise will be higher than the

in machining

To assess whether there is time for temperature gradients to even out, we compare the speed of the material with the speed of heat conduction.

To do this, estimate the thickness of the deformation zone, d , and an interaction time, t (the time h i l k h h h d f i ) Th di h h ld l i hi

p gaverage value calculated.

that material takes to pass through the deformation zone). The distance that heat could travel in this time is approximately , where is the thermal diffusivity. If d >> , temperature gradients will remain.

24

ta a ta

25

6. Microstructure Evolution in Wrought Alloy Processing

6.1 Microstructural characteristics of wrought alloys (Revision – mostly)

Recovery and recrystallisation

Central to metal forming are the mechanisms of recovery and recrystallisation. These may occur both during forming (“dynamic”) and after forming, during annealing (“static”).

These mechanisms fulfill several important purposes: - to maintain ductility (enabling large strains without cracking); - to reduce forming loads (dynamic softening balances work hardening;

annealing eliminates prior work hardening); - to control final grain structure.

Recrystallisation stems from the recovered subgrain structure, determined by: - prior deformation conditions (T, and ); - alloy composition.

There are two main sites for nucleation of recrystallisation: - grain boundaries, second phase particles.

Courtesy: Prof. J. Humphreys, University of Manchester

Grain boundary nucleation Larger subgrains at grain boundaries act as the nuclei for recrystallised grains.

Particle-stimulated nucleation (PSN) Wrought alloys contain fine-scale, hard, second phase particles and dispersoids (e.g. in Al alloys, intermetallic compounds of Al with Mn, Cr, Fe, Zr).

The dislocation density is greater around the hard particles, locally increasing the driving force for recovery and recrystallisation.

26

Grain size control by recrystallisation Recrystallisation requires a minimum strain level (typically 5% for cold deformation). Further strain leads to a decrease in recrystallised grain size (the number of nuclei increases). Similarly, there is a minimum temperature needed to trigger recrystallisation. This also falls with increasing strain, as it gets easier to start the process as the stored energy increases (LH figure below).

The recrystallised grain size has a complex dependence on: plastic strain (RH figure below) deformation strain-rate and temperature annealing temperature (for static recrystallisation)

Deformation processing is always inhomogeneous (due to geometric complexity, friction and heat transfer). Even in simple geometries such as flat strip rolling it is difficult to maintain uniform deformation across a rolled strip, and from one end of a coil to the other. In forging and extrusion deformation is very inhomogeneous. Hence different parts of the component will have different grain sizes, or may not recrystallise at all in some places. This can lead to problems with variable properties, anisotropic deformation behaviour, poor surface finish, localised corrosion etc.

Hot-rolling can allow deformation and recrystallisation in a single-stage process. Continuously cast steel is hot-rolled immediately after solidification, which produces a fine-grained homogeneous structure.

A subtle hidden benefit of recrystallisation relates to a reduction in impurity segregation (left by the prior casting process). As the new grain boundaries sweep through the material, they can drag some impurity atoms with them, partially levelling out concentration gradients – another mechanism for chemically homogenising cast material.

27

Heat-treatable Al alloys – age hardening

Aluminium dissolves up to 10% of Mg, Cu, Zn, Si, Li. Typical heat-treatable alloys contain at least two major alloy additions. The steps in age hardening are: (i) Solution heat treat, in the single phase region of the phase diagram. (ii) Quench to achieve a supersaturated solid solution. (iii) Age at room temperature (natural ageing) or at a temperature in the 150–250

oC range (artificial ageing).

Mechanism of age hardening The shape of the ageing curve results from the interaction of a number of effects: (i) rapid initial fine-scale precipitation from supersaturated solid solution. (ii) particle coarsening (i.e. steady decrease in the number of particles, with an increase in mean size and spacing), through one or more intermediate precipitates, eventually reaching the equilibrium phases. (iii) decrease in coherency (i.e. crystallographic matching) of the particle-matrix interface, as the particles coarsen and transform. (iv) transition from dislocations shearing the particles while they are small and coherent (the rising part of the curve), to dislocations bypassing the particles when they are well-spaced and incoherent (the falling part of the curve).

Artificial ageing: hardness and yield stress rise to a peak in about 5-24 hours (the "T6 temper") and then fall. Natural ageing: slow rise to a plateau hardness over 1-28 days (the "T4 temper").

YIELD STRESS

or

HARDNESS

Artificial ageing

Log (AGEING TIME)

Natural ageing

YIELD STRESS

or

HARDNESS

Artificial ageing

Log (AGEING TIME)

Natural ageing

Quench Quench

Age

Age

Solution treat

28

6.2 Case Study: Extrusion of heat-treatable Al alloys Hollow sections are made by building a mandrel into the die (held from behind by radial supports). The metal flow splits round the radial supports, then is forced together in a longitudinal friction weld beyond the mandrel, leaving an internal channel the same shape as the mandrel.

Multi-channel sections require many mandrels mounted sequentially along the die, generating complex 3D flow to produce the prismatic shape. Such dies can be very expensive (£100k).

(Source: Hydro Aluminium) Critical features of extrusion:

- die design - extrusion loads and speed, and optimisation of cooling schedules

- surface finish, product distortion and cracking - temperature and microstructure evolution (deformed grains, age hardening)

29

Temperature history and microstructural control in extrusion

T (oC)

TIME

DIRECT CHILL CAST INGOT

HOMOGENISATION

PREHEAT

EXTRUSION

QUENCH

AGE

Microstructural evolution: key ideas

- deformation and thermal histories are closely coupled - shaping and microstructure control are achieved simultaneously - the deformation/ageing stages inherit microstructure from upstream processes (e.g. casting and homogenisation) - deformation processing must leave the material with good properties for

downstream processing (e.g. heat treatment and welding), and for the product’s performance in service

Example of complex multi-process microstructure evolution

Prior processing history (casting and homogenisation) can have an important effect on extrusion and age hardening.

Al alloys contain intermetallic second phase particles and dispersoids (Al with Fe, Mn or Cr) formed during casting/homogenisation. Their size and number density depends on composition, and homogenization temperature & time.

Dispersoids are used to control recrystallised grain size (by PSN), but can act as nucleation sites for precipitation of coarse, non-hardening phases during quenching. This effectively removes solute from the supersaturated solution, and hence lower peak aged strength. The tendency for an alloy to suffer from this is known as quench sensitivity.

Quench sensitivity is a particular problem when it is difficult to impose a fast cooling rate – e.g. thick rolled plate, or extrusions of complex shape (which are more likely to distort).

30

log(TIME)

T

TE

T6 peak aged hardness

HARDNESS

log(COOLING RATE) (oC/s)1 2 5 10 20 50

Alloy 1

Alloy 2

C-curve for precipitation of coarse Hardness after artificial ageing, for (non-hardening) precipitates low dispersoid density (alloy 1) and high (alloy 2).

Example: quench sensitivity in high strength aerospace alloy 7010.

(Source: Alexis Deschamps, INP Grenoble)

Transmission electron micrographs of coarse precipitates nucleated on tiny spherical dispersoids during the quench. Each precipitate has “used up” the surrounding solute, as indicated by the absence of fine-scale precipitation near the precipitates during subsequent ageing. The final microstructure is a micro-composite of: normal peak-aged regions, and very soft precipitate-free regions. The net effect is an intermediate hardness, up to 50% below the peak-aged value.