Sketch a stress-strain curve for which the stress is calculated using the actual cross-sectional area rather than the original cross-sectional area. Explain why this curve differtypical stress-strain curve
To sketch a stress-strain curve where stress is calculated using the actual cross-sectional area instead of the original cross-sectional area, we need to understand the concept of stress and strain.
Stress (σ) is the force applied to a material divided by its cross-sectional area. It represents the internal resistance of a material to deformation due to an applied force. Stress is typically calculated using the original cross-sectional area (Ao) of the material before any deformation occurs.
Strain (ɛ) is the fractional change in length or shape of a material. It represents the amount of deformation that occurs due to the applied stress. Strain is typically calculated as the change in length (ΔL) divided by the original length (L) of the material.
In a typical stress-strain curve, the x-axis represents strain, while the y-axis represents stress. The curve starts from the origin, indicating no stress and no strain. As stress is applied to the material, it initially responds with elastic deformation, meaning it can return to its original shape when the applied force is removed. This is represented by the linear portion of the curve, known as the elastic region.
Once the material reaches its elastic limit, it enters the plastic region where it undergoes permanent deformation. The stress-strain curve increases non-linearly during this phase. Eventually, the material reaches its ultimate tensile strength, the maximum stress it can withstand before fracturing. Afterward, the stress decreases while the strain continues to increase until the material ultimately fails.
Now, if we calculate stress using the actual cross-sectional area (A) instead of the original cross-sectional area (Ao), the stress-strain curve would look different. When the material undergoes plastic deformation, the actual cross-sectional area decreases due to necking, where the material becomes narrower. This means that, as the material elongates, the area over which the force is distributed decreases. Consequently, this results in higher stress values being calculated for a given amount of strain compared to the curve using the original cross-sectional area.
Therefore, the stress-strain curve with stress calculated using the actual cross-sectional area would exhibit higher stress values throughout the plastic region and possibly a lower failure point compared to the curve using the original cross-sectional area. This is because considering the actual cross-sectional area accounts for the decreasing area during necking, resulting in a more accurate depiction of the material's ability to withstand deformation before failure.