14.5 Plastic Deformation, Yield Criteria & Load Estimation in Bulk Forming
Key Takeaways
- Plastic deformation and yield criteria, and load estimation for bulk forming, are named explicitly in the Casting, Forming and Joining bullet of the CIL Mechanical Paper-II syllabus.
- The Tresca criterion predicts yielding when the maximum shear stress reaches half the uniaxial yield stress, while the von Mises criterion uses distortion energy and is generally more accurate for ductile metals.
- In pure shear, Tresca gives the shear yield strength as half the tensile yield strength whereas von Mises gives it as the tensile yield divided by root three, so von Mises is the less conservative of the two.
- Hot working above the recrystallisation temperature needs far lower forces and produces no strain hardening, while cold working strengthens the metal but requires much higher loads.
Plastic Deformation
Metal forming works because metals deform plastically without fracture. The mechanism is slip — the movement of dislocations along close-packed planes in close-packed directions. Face-centred cubic metals such as aluminium and copper have many available slip systems and are highly formable; hexagonal close-packed metals such as magnesium and zinc have few and are difficult to form cold.
A critical property of plastic deformation is that it occurs at constant volume. Since the volume is unchanged,
in terms of true strains. This is why the Poisson's ratio of a metal in the fully plastic region is effectively 0.5, against about 0.3 in the elastic region.
True strain is the appropriate measure for large deformation:
and unlike engineering strain, true strains are additive across successive passes.
Strain hardening is described by the power law
where $K$ is the strength coefficient and $n$ the strain hardening exponent. A high $n$ distributes deformation uniformly and delays necking; indeed, in a tensile test, necking begins when $\epsilon = n$.
Yield Criteria
A uniaxial tensile test gives one number, the yield stress $\sigma_y$. Forming processes impose complex three-dimensional stress states, so a criterion is needed to predict when yielding begins.
Tresca (maximum shear stress) criterion
Yielding occurs when the maximum shear stress reaches the value it has at yield in simple tension:
where $\sigma_1$ and $\sigma_3$ are the largest and smallest principal stresses. The intermediate principal stress plays no part, which is both the criterion's simplicity and its limitation.
Von Mises (distortion energy) criterion
Yielding occurs when the distortion energy per unit volume reaches its value at uniaxial yield:
All three principal stresses participate. The criterion recognises that hydrostatic stress does not cause yielding — pure pressure changes volume, not shape — which is physically correct and confirmed experimentally: metals do not yield under uniform hydrostatic pressure however large.
Comparison
| Criterion | Shear yield strength | Relative prediction |
|---|---|---|
| Tresca | $k = 0.5,\sigma_y$ | More conservative (predicts yield sooner) |
| Von Mises | $k = \dfrac{\sigma_y}{\sqrt3} = 0.577,\sigma_y$ | More accurate for ductile metals |
The two criteria coincide in uniaxial tension and in balanced biaxial tension, and diverge most in pure shear, where they differ by about 15%. Plotted in principal stress space, the Tresca hexagon is inscribed within the von Mises ellipse.
For forming calculations the plane strain condition is common, and there the constrained yield stress rises:
This factor of 1.155 appears throughout rolling and forging load calculations.
Hot Working Versus Cold Working
The dividing line is the recrystallisation temperature, roughly $0.4,T_m$ on the absolute scale.
| Feature | Hot working (above $T_r$) | Cold working (below $T_r$) |
|---|---|---|
| Force required | Low | High |
| Strain hardening | None; recrystallisation removes it continuously | Significant; metal strengthens as it deforms |
| Achievable deformation | Very large | Limited before annealing needed |
| Surface finish | Poor; oxide scale forms | Good |
| Dimensional accuracy | Poor | Good |
| Grain structure | Refined, equiaxed | Elongated, directional |
| Residual stress | Negligible | Significant |
| Typical processes | Rolling of slabs, forging of billets, extrusion of steel | Cold rolling of sheet, wire drawing, coining |
Note that hot working is defined relative to the recrystallisation temperature, not in absolute terms. Lead recrystallises below room temperature, so working lead at 20 degrees Celsius is hot working; tungsten worked at 1000 degrees Celsius is still cold working. This is a classic objective item.
Load Estimation: Forging
For open-die upset forging of a cylindrical billet of diameter $d$ and height $h$ between flat dies, friction at the die faces produces the characteristic friction hill — pressure rises from the edge towards the centre. The average forging pressure is approximately
and the forging load is $F = p_{\text{avg}}\times\dfrac{\pi d^2}{4}$.
The implication is important: as the workpiece is flattened, $d$ grows and $h$ shrinks, so $d/h$ rises sharply and the required load escalates rapidly in the final stages. This is why forging presses are rated by the final thin section, not the starting billet.
Barrelling — the outward bulging of the sides — is caused by this same die friction restraining the ends. Good lubrication reduces both barrelling and load.
Load Estimation: Rolling
In flat rolling, the strip is drawn between rolls by friction. The maximum draft achievable in one pass is limited by the bite condition:
where $R$ is the roll radius. Two immediate consequences:
- Larger rolls take a bigger bite, which is why roughing mills use large rolls.
- Higher friction takes a bigger bite, which is why hot rolling is done dry or with minimal lubricant while cold rolling is heavily lubricated to reduce load, accepting smaller drafts.
The roll force is approximately
where $w$ is strip width, $L_p$ the projected contact length and $\bar\sigma$ the mean flow stress in plane strain. The torque per roll is roughly $F L_p/2$, and the power for two rolls is $2\pi N F L_p/60$.
The neutral point is where roll and strip surfaces move at the same speed. Before it the strip moves slower than the roll surface; after it, faster. The exit strip speed therefore exceeds the roll surface speed, an effect called forward slip.
Load Estimation: Extrusion
The extrusion ratio is $R_e = A_0/A_f$, and the ideal work per unit volume for homogeneous deformation is $\bar\sigma\ln R_e$. Including friction and redundant work empirically,
with $k$ typically 1.2 to 2.0.
| Type | Description |
|---|---|
| Direct (forward) | Ram and product move in the same direction; billet slides along the container, so friction is high and force falls as the billet shortens |
| Indirect (backward) | Die moves into a stationary billet; no relative sliding along the container, so force is lower and nearly constant |
| Hydrostatic | Billet surrounded by pressurised fluid; almost no container friction |
| Impact | High-speed cold extrusion of soft metals such as toothpaste tubes |
Load Estimation: Wire Drawing
Unlike extrusion, drawing pulls the material through the die, so the drawing stress cannot exceed the yield strength of the emerging wire, or it will simply stretch and break. This sets a hard limit:
For ideal frictionless drawing this gives $\ln(A_0/A_f) < 1$, so the maximum reduction in one pass is $1 - 1/e = 63%$ in theory. With friction and redundant work included, practical single-pass reductions are 20 to 30%, which is why wire is drawn through a series of progressively smaller dies with intermediate annealing.
That theoretical 63% limit is one of the most-asked single numbers in bulk forming.
According to the von Mises criterion, the shear yield strength of a ductile metal equals the tensile yield strength multiplied by:
Working lead at room temperature is classified as hot working because:
In flat rolling, the maximum draft achievable in a single pass is given by:
The theoretical maximum reduction in area in a single frictionless wire-drawing pass is approximately: