Stress Concentration at Structural Notch (Mechanical Engineering) Solution

Problem Overview

We analyze the geometric continuity, differentiability, and stress concentration characteristics of the V-notch profile:

at the notch vertex $x = 5$.

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Step-by-Step Mathematical Solution

Step 1: Prove Continuity at $x = 5$

Rewrite $y(x)$ as a piecewise function:

• Left-hand limit: $\lim_{x \to 5^-} (-x + 7) = -5 + 7 = 2$
• Right-hand limit: $\lim_{x \to 5^+} (x - 3) = 5 - 3 = 2$
• Function value: $y(5) = |5 - 5| + 2 = 2$

Since $\lim_{x \to 5^-} y(x) = \lim_{x \to 5^+} y(x) = y(5) = 2$, the bracket contour is continuous at $x = 5$.

Step 2: Test for Differentiability at $x = 5$

Compute the one-sided derivative limits using the difference quotient:

• Left-hand derivative:
  $$\lim_{h \to 0^-} \frac{y(5+h) - y(5)}{h} = \lim_{h \to 0^-} \frac{|(5+h) - 5| + 2 - 2}{h} = \lim_{h \to 0^-} \frac{|h|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$$
• Right-hand derivative:
  $$\lim_{h \to 0^+} \frac{y(5+h) - y(5)}{h} = \lim_{h \to 0^+} \frac{|h|}{h} = \lim_{h \to 0^+} \frac{h}{h} = +1$$

Because $\lim_{h \to 0^-} \frac{\Delta y}{h} = -1 \neq +1 = \lim_{h \to 0^+} \frac{\Delta y}{h}$, the two-sided limit does not exist. The function has a sharp corner and is non-differentiable at $x = 5$.

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Engineering Conclusion

In structural mechanics and elasticity theory, an abrupt geometric corner with zero radius of curvature ($r \to 0$) causes the theoretical stress concentration factor $K_t = 1 + 2\sqrt{a/r}$ to approach infinity. Under dynamic cyclic service loads, physical stress lines crowd at this non-differentiable vertex, inducing localized plastic deformation and micro-crack nucleation.

To eliminate this catastrophic failure mode, mechanical engineers replace sharp corners with tangent transition arcs or smooth polynomial curves (fillets), ensuring that the geometric derivative $y'(x)$ remains continuous ($C^1$) across the entire surface profile and distributing internal stresses uniformly.