◀ Back to Note

Tony's Intertwingled Memex • Reference Manual

Material Tensile Failure (Mechanical & Civil Engineering)

Formulas & Definitions

A materials testing laboratory subjects a high-tensile steel structural reinforcement rod to axial tensile loading. The internal tensile stress S(x)S(x)S(x) (in megapascals, MPa\text{MPa}MPa) is measured as a function of elongation displacement xxx (in millimeters), up to its critical physical fracture threshold at x=10 mmx = 10\text{ mm}x=10 mm:

S(x)={8x:0≤x<100:x≥10(The rod fractures completely)S(x) = \begin{cases} 8x & : 0 \le x < 10 \\ 0 & : x \ge 10 \quad \text{(The rod fractures completely)} \end{cases}S(x)={8x0​:0≤x<10:x≥10(The rod fractures completely)​

  1. Evaluate the left-sided limit lim⁡x→10−S(x)\lim_{x\to 10^-} S(x)limx→10−​S(x).
  2. Evaluate the right-sided limit lim⁡x→10+S(x)\lim_{x\to 10^+} S(x)limx→10+​S(x).
  3. Determine whether lim⁡x→10S(x)\lim_{x\to 10} S(x)limx→10​S(x) exists, and interpret the structural significance of the boundary behavior.