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 $\varepsilon = \ln(l/l<em>o) = \log</em>e(l/l<em>o)$, where $l$ is the final length and $l</em>o$ is the initial length; 2.718.
• Relationship between true and engineering values:
  • $\sigma = \sigma<em>o(1+e)$, where $\sigma</em>o$ 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: $A \cdot l = A<em>o \cdot l</em>o$, so $A = A<em>o (l</em>o / l)$.
• True stress $\sigma = F/A = (F/A<em>o)(l/l</em>o) = \sigma<em>o(l/l</em>o)$.
• Since engineering strain $e = (l - l<em>o)/l</em>o = (l/l<em>o) - 1$, then $l/l</em>o = 1 + e$.
• Therefore, true stress $\sigma = \sigma_o(1+e)$.
• Infinitesimal length increase $dl$ corresponds to a strain increment $d\varepsilon = dl/l$.
• True strain $\varepsilon = \int<em>{l</em>o}^{l} d\varepsilon = \int<em>{l</em>o}^{l} dl/l = \ln l - \ln l<em>o = \ln(l/l</em>o) = \ln(1+e)$.

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, $e<em>{UTS}$, is used to find $K$ as $\sigma</em>{UTS} = K \cdot e<em>{UTS}^n$, where $\sigma</em>{UTS}$ is true stress at UTS.

Poisson’s Ratio, ν

• Lateral true strain: $\epsilon<em>t = \ln(D/D</em>o)$, where $D_o$ is initial diameter and $D$ is final diameter.
• Longitudinal true strain: $\epsilon<em>l = \ln(l/l</em>o)$
• Poisson’s ratio: $\nu = -\epsilon<em>t / \epsilon</em>l$
• 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 $\sigma<em>2/\sigma</em>1 = (\dot{\epsilon<em>2}/\dot{\epsilon</em>1})^{0.025} = (100/1)^{0.025} = 1.122$.
• 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 $0.4T<em>m - 0.5T</em>m$ 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 > 0.5$T_{melt}$): Recrystallization is simultaneous with plastic deformation; true stress ≈ Y (constant mean yield stress).
• Cold-forming (T < 0.3$T<em>{melt}$): Work-hardening occurs; $\sigma</em>{true} = K \cdot \epsilon_{true}^n$