Plastic deformation, yield criteria and hot vs cold working
Plastic flow, true strain, Hollomon flow stress and average flow stress, Tresca and von Mises criteria, plane strain, and hot, warm and cold working.
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Why it matters
Every forming calculation — forging load, rolling force, extrusion pressure, drawing stress — needs two inputs: when the metal starts to flow (a yield criterion) and how hard it is to keep it flowing (the flow stress at the working temperature and strain). Choosing hot or cold working sets both, and also decides the finish, accuracy and properties of the part.
Key ideas
Plastic deformation. Beyond the elastic limit, metal deforms permanently by slip — the movement of dislocations on close-packed planes. Plastic flow is essentially at constant volume (incompressible), so the three principal plastic strains add to zero: ε₁ + ε₂ + ε₃ = 0. Hydrostatic pressure does not cause yielding; only the deviatoric (shape-changing) part of the stress does. That is why metals can be extruded under enormous pressures without failing, and why compressive hydrostatic stress improves workability.
True stress and true strain. In forming, strains are large, so engineering strain is misleading. True strain ε = ln(L/L₀) (or ln(h₀/h) for compression) is additive over successive steps; true stress σ = F/A uses the current area.
Flow stress and strain hardening. In cold working the flow stress rises with strain, usually modelled by the Hollomon law σ_f = K·εⁿ, with K the strength coefficient and n the strain-hardening exponent (about 0.1–0.5 for common metals). For a process whose strain goes from 0 to ε, the average flow stress is used in force formulas.
Yield criteria decide when a multiaxial stress state causes yielding, given the uniaxial yield stress σ_y.
- Tresca (maximum shear stress): yielding when the largest principal-stress difference reaches σ_y: σ_max − σ_min = σ_y. Shear yield stress k = σ_y/2. Simple, slightly conservative, used for quick forming estimates.
- von Mises (distortion energy): yielding when the equivalent stress reaches σ_y. Shear yield stress k = σ_y/√3 = 0.577σ_y. It agrees better with experiments on ductile metals; the two criteria differ by at most about 15%.
- Plane strain (wide strip rolling, forging of long bars) — one dimension cannot change. Under von Mises the yield stress in plane strain compression is 2σ_y/√3 = 1.155σ_y (Tresca gives σ_y). This 1.155 factor appears in rolling and forging formulas.
Hot, warm and cold working. The dividing line is the recrystallisation temperature, roughly 0.4–0.5 of the absolute melting temperature T_m (homologous temperature T/T_m).
- Cold working (below about 0.3 T_m) — strain hardening raises strength and hardness and lowers ductility; good finish and close tolerances; no scale; higher forces, limited deformation before annealing is needed; grains elongated and residual stresses left behind.
- Warm working (about 0.3–0.5 T_m) — lower forces than cold, better accuracy than hot.
- Hot working (above about 0.6 T_m) — recrystallisation keeps pace with deformation, so there is essentially no strain hardening; large deformations at low force; porosity closes and cast structure is refined; but scale, poorer finish and tolerance, and decarburisation in steels. Flow stress now depends on strain rate: σ_f = C·ε̇ᵐ, with m the strain-rate sensitivity (small at room temperature, about 0.05–0.4 hot; very high in superplastic alloys).
- Lead and tin recrystallise near room temperature, so working them at room temperature is hot working; tungsten worked at 1100 °C is still cold working.
Annealing after cold work restores ductility through recovery, recrystallisation and grain growth.
Formulas
ε = ln(L/L₀) or ε = ln(h₀/h) — true strain (dimensionless).
σ_f = K·εⁿ — flow stress in cold working (MPa); K = strength coefficient (MPa), n = strain-hardening exponent.
Ȳ = K·εⁿ / (1 + n) — average flow stress over a strain from 0 to ε (MPa).
σ_f = C·ε̇ᵐ — hot-working flow stress (MPa); C = strength constant (MPa), ε̇ = strain rate (s⁻¹), m = strain-rate sensitivity.
Tresca: σ_max − σ_min = σ_y, k = σ_y / 2.
von Mises: σ_e = √(½[(σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²]) = σ_y, k = σ_y / √3.
Plane stress (σ₃ = 0): σ_e = √(σ₁² − σ₁σ₂ + σ₂²).
Plane-strain yield stress (von Mises) = (2/√3)·σ_y = 1.155·σ_y.
Worked examples
Example 1 (standard — flow stress). A cylinder is upset cold from 50 mm to 30 mm height. The metal has K = 500 MPa and n = 0.2. Find the true strain, the final flow stress and the average flow stress.
ε = ln(50/30) = 0.511.σ_f = 500 × 0.511^0.2 = 437 MPa.Ȳ = 437 / 1.2 = 364 MPa.
Answer: ε = 0.511, σ_f ≈ 437 MPa, Ȳ ≈ 364 MPa.
Example 2 (GATE level — criteria compared). A point in a ductile part has σ₁ = 180 MPa, σ₂ = −60 MPa, σ₃ = 0. The uniaxial yield stress is 220 MPa. Does it yield by Tresca? By von Mises?
- Tresca: σ_max − σ_min = 180 − (−60) = 240 MPa. Since 240 > 220, yielding is predicted.
- von Mises:
σ_e = √(180² − 180 × (−60) + (−60)²) = √(32 400 + 10 800 + 3600) = √46 800 = 216.3 MPa. Since 216.3 < 220, no yielding.
Answer: Tresca predicts yield, von Mises does not — the region between the two yield loci, where Tresca is more conservative. Note that σ_min here is −60 MPa, not 0.
Example 3 (hot working). For a steel at 1100 °C, C = 100 MPa and m = 0.1. Find the flow stress at strain rates of 1 s⁻¹ and 10 s⁻¹.
σ_f(1) = 100 × 1^0.1 = 100 MPa.σ_f(10) = 100 × 10^0.1 = 125.9 MPa.
Answer: 100 MPa and ≈126 MPa — a tenfold increase in speed raises the hot flow stress by about 26%.
Common mistakes
- Taking σ_min as zero in Tresca when one principal stress is negative. Use the algebraically largest minus the algebraically smallest of all three, including σ₃ = 0.
- Using engineering strain in K·εⁿ. Use true strain.
- Using the final flow stress where the average is needed (work, energy) or vice versa (peak force at the end of a stroke).
- Defining hot working by an absolute temperature such as "above 500 °C". It is relative to the metal's recrystallisation temperature.
- Forgetting the 1.155 plane-strain factor in rolling and forging of wide strips when the question uses von Mises.
- Thinking hydrostatic stress causes yielding; it does not.
For GATE PI
Expect numericals that ask for true strain, average flow stress from the Hollomon law, yield checks by Tresca and von Mises (often both, as above), shear yield stress under each criterion, and hot-working flow stress from C and m. One-mark questions test cold vs hot working effects, recrystallisation temperature, and which criterion is more conservative. Practise plane-stress von Mises quickly and watch the sign of σ₂.
Quick check
- Shear yield stress by von Mises for σ_y = 300 MPa?
- A bar is stretched from 100 mm to 150 mm. True strain?
- Is lead rolled at room temperature cold or hot worked?
- Principal stresses 200, 100, 0 MPa. von Mises equivalent stress?
- Which criterion gives the higher shear yield stress for the same σ_y?
Answers: 1. 300/√3 ≈ 173 MPa. 2. ln 1.5 = 0.405. 3. Hot worked (recrystallises at room temperature). 4. √(200² − 200 × 100 + 100²) ≈ 173 MPa. 5. von Mises (0.577σ_y vs 0.5σ_y).
Interview questions
All Casting, Forming and Joining interview questionsTry answering each one aloud before you open it.
1.What is plastic deformation in the context of materials engineering?Concept
Plastic deformation refers to the permanent change in shape or size of a material when subjected to a stress that exceeds its elastic limit. Unlike elastic deformation, which is reversible, plastic deformation remains even after the stress is removed. It occurs due to the movement of dislocations within the material's crystal structure.
2.Explain the concept of yield criteria in materials science.Concept
Yield criteria are mathematical models that predict the onset of plastic deformation in materials under various states of stress. Common yield criteria include the von Mises and Tresca criteria. The von Mises criterion is based on the distortion energy theory, while the Tresca criterion is based on the maximum shear stress theory. These criteria help in determining the yield point of materials under complex loading conditions.
3.What is the difference between hot working and cold working processes?Concept
Hot working involves deforming materials at temperatures above their recrystallization temperature, which allows for easier shaping and prevents strain hardening. Cold working, on the other hand, is performed at temperatures below the recrystallization temperature, leading to strain hardening and increased strength but reduced ductility. Hot working is typically used for large deformations, while cold working is used for precision and improved surface finish.
4.Why is the von Mises yield criterion often preferred over the Tresca criterion in engineering applications?Application
The von Mises yield criterion is often preferred because it provides a more accurate prediction of yielding for ductile materials under complex loading conditions. It considers the distortion energy in the material, which is a more realistic representation of the yielding process compared to the maximum shear stress approach of the Tresca criterion. Additionally, the von Mises criterion results in a smooth yield surface, which is mathematically convenient for analysis.
5.What happens to the mechanical properties of a metal when it undergoes cold working?Application
When a metal undergoes cold working, its mechanical properties change significantly. The process increases the strength and hardness of the metal due to strain hardening. However, it also reduces the ductility, making the metal more brittle. The grain structure becomes elongated, and dislocation density increases, which contributes to these changes.
6.In what scenarios would hot working be preferred over cold working?Application
Hot working is preferred when large deformations are required, as it allows for easier shaping of the material without the risk of fracture. It is also used when the material needs to be processed quickly and efficiently, as the elevated temperatures reduce the flow stress. Additionally, hot working is suitable for materials that are difficult to work with at room temperature, such as high-strength alloys.
7.Calculate the von Mises equivalent stress for principal stresses σ₁ = 200 MPa, σ₂ = 100 MPa and σ₃ = 0. When will the material yield?Numerical
σ_e = √(½[(σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²]) = √(½[100² + 100² + 200²]) = √30 000 = 173.2 MPa. The material yields if its uniaxial yield stress is 173.2 MPa or less. For comparison, Tresca uses σ_max − σ_min = 200 − 0 = 200 MPa, so it is more conservative here.
8.If a metal is cold worked to 50% reduction in area, how does this affect its yield strength?Application
Cold working a metal to a 50% reduction in area significantly increases its yield strength due to strain hardening. The dislocation density within the metal increases, which impedes further movement of dislocations and thus increases the yield strength. The exact increase in yield strength depends on the material and its initial properties, but it is generally substantial.
9.Explain why annealing is often performed after cold working.Application
Annealing is performed after cold working to relieve internal stresses, reduce hardness, and restore ductility. The process involves heating the material to a specific temperature and then cooling it slowly. This allows for recrystallization, where new grains form without the dislocations that were introduced during cold working, thus improving the material's ductility and reducing brittleness.
10.A cylindrical rod is subjected to a tensile stress of 150 MPa. If the yield strength of the material is 200 MPa, will the rod undergo plastic deformation?Numerical
No, the rod will not undergo plastic deformation because the applied tensile stress of 150 MPa is below the yield strength of the material, which is 200 MPa. Plastic deformation occurs only when the applied stress exceeds the yield strength. In this case, the rod will experience elastic deformation, which is reversible upon removal of the stress.
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