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 | $K_c$ | 53 MPa$\sqrt{\text{m}}$ |
| Yield strength | $\sigma_y$ | 950 MPa |
| Smallest detectable surface crack | $a$ | 1.0 mm |
| Configuration factor | $Y$ | 1.12 (standard surface/edge-crack factor — not given) |
Find. Whether the plate reaches general yield or fractures first, at the detection-limit crack size.
Approach. Compute the nominal stress $\sigma_f$ that would drive the detection-limit crack to $K=K_c$, and compare it directly to $\sigma_y$: whichever stress is lower is reached first as load increases.
Conclusion. A plate containing a crack at the ultrasonic detection limit will fail by fast (brittle-type) fracture at about 844 MPa, before general yielding can occur at 950 MPa — even though the parent steel itself is a normally ductile, high-strength material. The inspection limit is therefore not automatically "safe": any undetected crack up to 1.0 mm still leaves roughly an 11% stress margin below the nominal yield strength unusable, because fracture intervenes first.
| Quantity | Value |
|---|---|
| Fracture stress at $a=1.0$ mm, $\sigma_f$ | 844.3 MPa |
| Yield strength, $\sigma_y$ | 950 MPa |
| Governing mode | Fast fracture (before general yield) |
Body-centred-cubic ferritic/pearlitic structural steels exhibit a ductile-to-brittle transition temperature (DBTT): the Peierls (lattice friction) stress that resists dislocation glide in the BCC lattice is strongly thermally activated, so as temperature falls the stress needed to nucleate and sustain plastic (dislocation glide) deformation rises sharply, while the fracture stress governed by $K_{Ic}$ changes far more slowly. Below the DBTT, the rapidly rising "yield-like" flow stress can exceed the stress needed for brittle cleavage fracture before enough plastic deformation can occur to blunt a crack tip or a stress-concentrating flaw (weld defect, notch, cold-formed corner) — the fracture toughness itself also drops substantially in this low-temperature regime. Winter service temperatures push large welded steel structures (thick sections favouring plane-strain constraint, with unavoidable welding flaws and residual/thermal stresses) toward or below this transition, so a crack that would have been safely arrested by local plastic flow in summer instead propagates catastrophically in winter — the historical Liberty-ship brittle-fracture failures are the classic illustration of exactly this effect.
General plastic deformation of a polycrystal requires, by the von Mises criterion, at least five independent slip systems operating at comparable stress so that neighbouring grains can accommodate an arbitrary shape change without cracking at their boundaries. FCC metals have 12 easy $\{111\}\langle110\rangle$ slip systems and BCC metals have a large number of $\{110\}/\{112\}/\{123\}\langle111\rangle$ systems, both comfortably exceeding five, so both crystal structures can deform homogeneously and ductilely as polycrystals. HCP metals, in contrast, have only 3 independent easy slip systems on the single basal plane $(0001)\langle11\bar{2}0\rangle$; non-basal systems (prismatic, pyramidal) exist but have a critical resolved shear stress many times higher, so they contribute little at ordinary temperatures. With fewer than five easily activated systems, adjacent HCP grains cannot mutually accommodate an arbitrary strain by slip alone, so incompatibility stresses build up at grain boundaries and are relieved by twinning and, ultimately, by cracking rather than by further slip — making polycrystalline HCP metals (Mg, Zn, Be, Ti at room temperature) markedly more brittle than FCC or BCC metals of comparable purity.
