Bridge Truss Gusset Plate Fatigue (Mechanical & Civil Engineering)

During a structural health inspection of a steel bridge truss, engineers monitor the stress intensity factor $K$ (in $\text{MPa}\cdot\sqrt{\text{m}}$ ) around a critical micro-crack on a primary gusset plate. The intensity factor depends on the applied tension stress $\sigma$ (in $\text{MPa}$ ) and the physical length of the crack $a$ (in meters), modeled by:

K(\sigma, a) = 1.5\sigma \sqrt{a} = 1.5\sigma a^{0.5}

As heavy morning commuter traffic moves onto the bridge, the load-induced stress peaks. Simultaneously, due to structural fatigue, the crack lengthens dynamically over time $t$ (in minutes) according to the parameters:

$\sigma(t) = 200 + 10t^2$
$a(t) = 0.04 + 0.002t$

Compute the baseline tensile stress $\sigma(2)$ and crack length $a(2)$ at $t = 2\text{ minutes}$ .
State the multivariable chain rule formula for the total time rate of change $\frac{dK}{dt}$ .
Calculate the instantaneous rate of change of the stress intensity factor $\frac{dK}{dt}$ at $t = 2\text{ minutes}$ .
In fracture mechanics, crack propagation becomes unstable when $K \ge K_{Ic}$ (fracture toughness). Explain why structural health monitoring systems track the rate $\frac{dK}{dt}$ rather than just the instantaneous magnitude $K$ .