Longitudinal strain can indeed be negative, and this concept is crucial when analyzing materials under thermal expansion or contraction. In your scenario with the steel rod, the key lies in understanding how temperature changes affect the material's dimensions, especially when it is constrained at both ends.
Understanding Longitudinal Strain
Longitudinal strain is defined as the change in length of a material divided by its original length. Mathematically, it can be expressed as:
Longitudinal Strain (ε) = ΔL / L₀
Where:
- ΔL = Change in length
- L₀ = Original length
Thermal Expansion Basics
When a material is heated, it typically expands. The coefficient of linear expansion (α) quantifies how much a material expands per degree of temperature change. For steel, this value is generally around 11 x 10-6 /°C. If the temperature of the steel rod increases from 20°C to 50°C, the change in temperature (ΔT) is:
ΔT = 50°C - 20°C = 30°C
Calculating the Expected Change in Length
The change in length (ΔL) due to thermal expansion can be calculated using the formula:
ΔL = L₀ * α * ΔT
Since the rod is clamped at both ends, it cannot expand freely. This restriction leads to internal stresses and strains. The longitudinal strain developed in the rod can be expressed as:
ε = -α * ΔT
The negative sign indicates that the strain is compressive. In this case, the rod is trying to expand due to the increase in temperature, but because it is fixed at both ends, it cannot do so, resulting in a negative strain value.
Clarifying the Discrepancy in Your Answer
If your calculation yielded a positive strain, it likely did not account for the constraints imposed by the clamping. The book's answer of negative strain is correct because it reflects the compressive nature of the strain when the material is unable to expand freely. In essence, the rod is attempting to elongate, but the fixed ends prevent this, leading to a compressive strain.
Example for Better Understanding
Consider a rubber band that you stretch between two fixed points. If you heat the rubber band, it wants to expand. However, if you hold it tightly at both ends, it cannot stretch, and instead, it experiences a compressive force. This is similar to what happens with the steel rod in your problem.
Final Thoughts
In summary, when a material is constrained and subjected to thermal expansion, the longitudinal strain can indeed be negative. This reflects the internal stresses that arise from the inability of the material to expand freely. Always remember to consider the constraints of the system when analyzing thermal effects on materials.
