True Stress-Strain & Strain Rate

True Stress-Strain & Strain Rate

Necking and Failure

• After the Ultimate Tensile Strength (UTS), necking occurs, concentrating plastic deformation.
• Fracture happens within this neck.
• In stress-strain graphs, stress is usually calculated as applied load divided by the initial cross-sectional area.
• However, the true stress is higher due to the decreasing cross-sectional area during the tensile test.
• The maximum stress on the curve is the material's tensile strength or UTS.

True Stress & True Strain

• Engineering stress is F/A_o (applied load / initial cross-sectional area).
• Engineering strain is \Delta l/l_o (change in length / initial length).
• True stress and true strain consider the instantaneous cross-section area and length during deformation.
• True stress is the real stress experienced, and true strain is the real strain.
• During a tensile test the cross-section area decreases as its length increases
• Thus, real or true tensile stress is greater and true strain is lower than the values calculated from initial dimensions.

Calculating True Stress & True Strain

• True stress is given by \sigma = F/A, where A is the instantaneous area.
• True strain is given by \varepsilon = \ln(l/lo) = \loge(l/lo), where l is the final length and lo is the initial length; 2.718.
• Relationship between true and engineering values:
  • \sigma = \sigmao(1+e), where \sigmao is engineering stress and e is engineering strain.
  • \varepsilon = \ln(1+e)

Proofs of True- & Engineering- Stress & Strain Relationships

• During plastic deformation, volume remains constant: A \cdot l = Ao \cdot lo, so A = Ao (lo / l).
• True stress \sigma = F/A = (F/Ao)(l/lo) = \sigmao(l/lo).
• Since engineering strain e = (l - lo)/lo = (l/lo) - 1, then l/lo = 1 + e.
• Therefore, true stress \sigma = \sigma_o(1+e).
• Infinitesimal length increase dl corresponds to a strain increment d\varepsilon = dl/l.
• True strain \varepsilon = \int{lo}^{l} d\varepsilon = \int{lo}^{l} dl/l = \ln l - \ln lo = \ln(l/lo) = \ln(1+e).

True Stress vs. True Strain: Why So Useful

• For many metals, the relationship is \sigma = K \cdot \varepsilon^n, where K is a constant and n is the strain-hardening exponent (typically 0.1 - 0.6).
• This formula allows engineers to simulate plastic deformation processes.
• K and n can be found from experimental tensile tests by calculating true stress and true strain at the UTS.
• The true strain at UTS, e{UTS}, is used to find K as \sigma{UTS} = K \cdot e{UTS}^n, where \sigma{UTS} is true stress at UTS.

Poisson’s Ratio, ν

• Lateral true strain: \epsilont = \ln(D/Do), where D_o is initial diameter and D is final diameter.
• Longitudinal true strain: \epsilonl = \ln(l/lo)
• Poisson’s ratio: \nu = -\epsilont / \epsilonl
• For elastic deformation, \nu is typically 0.25 - 0.3.
• For plastic deformation, \nu is approximately 0.5 for any metal.

Effect of Strain-Rate on Stress-Strain Behaviour

• Strain rate affects flow stress (yield stress) according to \sigma \propto (\dot{\epsilon})^m, where typically m < 0.025.
• Example: If m = 0.025, a 100x increase in strain rate increases flow stress by 12.2% because \sigma2/\sigma1 = (\dot{\epsilon2}/\dot{\epsilon1})^{0.025} = (100/1)^{0.025} = 1.122.
• The effect of strain rate can be very dramatic for some polymers like polyethylene.

Effect of Temperature

• Flow stress is the stress required to continue plastic deformation, generally increasing with strain.
• Work-hardening rate is d\sigma/d\epsilon, the gradient of the true stress vs. true strain curve, generally decreasing with strain.
• Most metals work-harden and become stronger during plastic deformation at near room temperature, specifically at T \leq 0.3T_{melt} (in Kelvin); this is called cold-working; for many metals: \sigma = K \cdot \epsilon^n where K is a constant and n is the strain-hardening exponent (typically 0.1 - 0.6).

Effect of Temperature: Cold and Hot Working

• At T < 0.3T_{melt}, cold-working increases strength and hardness but decreases ductility.
• Ductility can be restored by annealing (heating to 0.4Tm - 0.5Tm in Kelvin), causing recrystallization and removing internal stresses.
• At T > 0.5T_{melt}, metals are soft and do not work harden; hot-working involves simultaneous recrystallization during deformation.

Stress-Strain Behaviour

• Hot-forming (T > 0.5T_{melt}): Recrystallization is simultaneous with plastic deformation; true stress ≈ Y (constant mean yield stress).
• Cold-forming (T < 0.3T{melt}): Work-hardening occurs; \sigma{true} = K \cdot \epsilon_{true}^n