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LEFM — singular tip field
KI = 0
Near-tip stress $\sigma_{\theta\theta}(r)$ along the ligament
$K_I/\sqrt{2\pi r}$
Finite-plate (bounded)
Probe
$K_I$:
0.0 MPa√mm
$\sigma_{\mathrm{tip}}(r)$:
0.0 MPa
Amplification $\sigma/\bar{\sigma}$:
—
Crack $2a$:
40 mm
Concept:
σ → ∞ as r → 0
Log–log check: $\log\sigma$ vs $\log r$ (slope $\approx -1/2$)
Williams Mode I idea
Ahead of the tip on the crack plane ($\theta=0$):
$$\sigma_{yy}(r,0)=\frac{K_I}{\sqrt{2\pi r}}+\mathcal{O}(1)$$
As $r\to 0$, stress unbounded — the LEFM singularity. $K_I=\bar{\sigma}\sqrt{\pi a}\,Y(a/W)$ sets intensity; geometry factor $Y$ ≈ 1 for small $a/W$.
Specimen
- Plate: square side $W=L=100$ mm
- Crack: centered, length $2a$, Mode I opening
- BCs: bottom supported · top remote $\bar{\sigma}_{yy}$
- Heat colors = local $\sigma_{yy}$ intensity (dark → hot near tips)
What to notice
- Move the probe toward the tip — stress rises without bound in LEFM
- Log–log plot stays near slope −½ in the K-dominant zone
- Larger $a$ or $\bar{\sigma}$ raises $K_I$ and the whole near-tip field
- Real materials cut off the singularity with a process / plastic zone