The question refers to the true stress-strain relationship of materials undergoing plastic deformation, specifically focusing on the point where necking begins. The given relationship is expressed as \(\rm \sigma _ T= K \varepsilon _T^n\), where \(\sigma_T\) is the true stress, \(\varepsilon_T\) is the true strain, \(K\) is the strength coefficient, and \(n\) is the strain hardening exponent. We need to find the value of the strain hardening exponent \(n\) for low carbon steel.

Stress-Strain Relation in Plastic Deformation

During plastic deformation, materials exhibit a behavior described by the true stress-strain curve. Unlike engineering stress and strain, true stress and true strain are calculated based on the instantaneous cross-sectional area and length of the material during deformation. The power law relationship \(\rm \sigma _ T= K \varepsilon _T^n\) is widely used to approximate the plastic behavior of many metals, including steel, particularly at strains beyond the yield point and up to the onset of necking.

• True Stress (\(\sigma_T\)): Instantaneous load divided by the instantaneous cross-sectional area.
• True Strain (\(\varepsilon_T\)): Sum of incremental strains, or \( \ln(L/L_0) \), where \(L\) is instantaneous length and \(L_0\) is original length.
• Strength Coefficient (K): A material constant representing the stress at a true strain of 1.0 (if n=1). Its units are typically MPa or psi.
• Strain Hardening Exponent (n): A material property that indicates how much a material can strengthen with plastic deformation. A higher \(n\) value signifies a greater ability to undergo uniform plastic deformation before localization (necking) occurs.

Necking Phenomenon and its Onset

Necking is a phenomenon that occurs during the tensile testing of ductile materials. It refers to the localized reduction in the cross-sectional area of the specimen, usually initiating at the point of maximum true stress. Once necking begins, the deformation becomes concentrated in this localized region, leading to fracture. The true stress-strain relationship is crucial for understanding this phenomenon.

The condition for the onset of necking is when the rate of strain hardening becomes equal to the true stress. Mathematically, this is expressed as:

\[ \frac{d\sigma_T}{d\varepsilon_T} = \sigma_T \]

Let's use the given true stress-strain relation \(\rm \sigma _ T= K \varepsilon _T^n\) to find the true strain at which necking begins:

1. Differentiate \(\sigma_T\) with respect to \(\varepsilon_T\): \[ \frac{d\sigma_T}{d\varepsilon_T} = \frac{d}{d\varepsilon_T} (K \varepsilon_T^n) = n K \varepsilon_T^{n-1} \]
2. Set this equal to \(\sigma_T\) for the necking condition: \[ n K \varepsilon_T^{n-1} = K \varepsilon_T^n \]
3. Cancel out \(K\) (assuming \(K \neq 0\)): \[ n \varepsilon_T^{n-1} = \varepsilon_T^n \]
4. Divide both sides by \(\varepsilon_T^{n-1}\) (assuming \(\varepsilon_T \neq 0\)): \[ n = \frac{\varepsilon_T^n}{\varepsilon_T^{n-1}} \] \[ n = \varepsilon_T^{n - (n-1)} \] \[ n = \varepsilon_T^1 \] \[ n = \varepsilon_T \]

Therefore, at the point where necking begins, the true strain \(\varepsilon_T\) is numerically equal to the strain hardening exponent \(n\).

Low Carbon Steel Properties

The strain hardening exponent \(n\) is an important mechanical property that varies for different materials. For many metals, including various types of steel, the value of \(n\) typically ranges from 0.1 to 0.5. A higher \(n\) value implies greater ductility and formability, as the material can withstand more uniform plastic deformation before local instability (necking) occurs.

For low carbon steel, a common and widely accepted approximate value for the strain hardening exponent \(n\) is 0.21. This value reflects the material's moderate work-hardening capacity and its good formability.

Based on empirical data and material science literature, the value of \(n\) for low carbon steel is generally found to be around 0.21.