21-Mat-A4 Deformation Behaviour and Properties of Materials · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, May 2017 — 12-Mtl-A4, Mechanical Behaviour and Fracture of Materials. Three hours, closed book, one approved non-communicating calculator. Eight questions of 20 marks each; the rubric marks only the first five questions as they appear in the answer book. All eight are solved here. This sitting's own printed content is fracture mechanics, fatigue crack growth, creep, toughness, strengthening, composites and plastic instability.
Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Given.
| Quantity | Symbol | Value |
|---|---|---|
| Fracture toughness | $K_c$ | 25 MPa$\sqrt{\text{m}}$ |
| Cyclic (range) stress | $\Delta\sigma$ | 100 MPa |
| Initial crack length | $a_i$ | 2.0 mm |
| Paris-law coefficient | $A$ | $1\times10^{-12}$ MPa$^{-3}$m$^{-1/2}$ |
| Paris-law exponent | $n$ | 3 |
| Configuration factor | $Y$ | 1 (large sheet — through-crack idealisation) |
Find. The number of cycles to failure, $N_f$.
Approach. First find the critical crack size $a_c$ at which $K=K_c$ under the applied $\Delta\sigma$ (treating the peak of the cycle as the fracture-driving stress), then integrate the Paris law from $a_i$ to $a_c$ for $N_f$.
| Quantity | Value |
|---|---|
| Critical crack length, $a_c$ | 19.9 mm |
| Fatigue life, $N_f$ | $5.49\times10^{6}$ cycles |
Mechanism 1 — Fatigue. Under stresses well below $\sigma_y$ but applied cyclically, local plastic strain still occurs at microscopic stress concentrators (inclusions, surface scratches, grain-boundary triple points) even though the bulk nominal stress never reaches yield. Repeated cyclic slip produces persistent slip bands that extrude and intrude at the free surface, nucleating a microcrack after a characteristic number of cycles. The crack then grows sub-critically according to the Paris law of part (a) until it reaches the size at which $K=K_{Ic}$, at which point sudden fast fracture occurs — all at a nominal stress that, examined on the ordinary stress-strain curve alone, looks perfectly safe.
Mechanism 2 — Flaw-driven brittle (fast) fracture, governed by LEFM. If the component already contains a pre-existing sharp crack or crack-like defect (a weld flaw, a forging lap, a fatigue crack from mechanism 1, hydrogen-induced cracking, or stress-corrosion cracking), the local crack-tip stress field can reach the critical intensity $K_{Ic}$ while the remote/nominal stress is still below $\sigma_y$, exactly as demonstrated numerically in Question 1(a) and Question 5(a) of this paper. The larger the flaw and the lower the toughness (favoured by low temperature, high strain rate, and thick sections promoting plane-strain constraint), the lower the nominal stress at which this fast, largely energy-releasing fracture occurs — with essentially no warning plastic deformation.
