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21-Mat-A5 Phase Transformations and Thermal Treatment · December 2017

Question 7 of 8: Ductile vs. Cyclic Fracture Micromechanisms; the Necking (Plastic Instability) Condition

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Exams, December 2017 — 10-Met-A5, Mechanical Behaviour and Fracture of Materials. Three hours, closed book, any Casio- or Sharp-approved calculator permitted. Eight questions of 20 marks each; the rubric states that five questions constitute a complete paper and that only the first five appearing in the answer book are marked. All eight are answered here, because this set is a study resource rather than an exam script. Several sub-parts explicitly call for an essay-format answer, and the rubric rewards clarity and organisation, so those answers are written as structured prose rather than as note form.

Note on the exam title

Nothing on the paper is a phase-transformation or heat-treatment question in the TTT/CCT, hardenability or tempering sense; the syllabus actually examined is dislocation theory, slip and twinning, strengthening mechanisms, creep, fatigue, toughness and fracture mechanics, and deformation processing.

Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:

Question 7: Ductile vs. Cyclic Fracture Micromechanisms; the Necking (Plastic Instability) Condition (20 marks)

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.

7.1 — (a) Fracture micromechanisms: ductile static overload vs. cyclic loading

The same ductile metal fails by physically different micromechanisms, and therefore leaves visibly different fracture surfaces, depending on whether the load is applied once (statically, to overload) or many times (cyclically).

Static ductile overload proceeds by microvoid coalescence: as the stress rises toward the ultimate tensile strength, voids nucleate at second-phase particles and inclusions (by particle cracking or interface decohesion), grow under the local triaxial stress state built up ahead of a developing neck, and finally link up by internal necking of the ligaments between them. The macroscopic signature is the classic cup-and-cone fracture — a flat, dimpled central region (voids nucleated and grew under high triaxiality) surrounded by a 45$^\circ$ shear lip (final separation under plane-stress shear) — and, at the microscale, a fracture surface entirely covered in equiaxed dimples, each one the signature of a single coalesced void, sized and shaped by the local particle spacing and stress state. The whole process consumes one monotonically increasing load history and requires substantial macroscopic plastic strain (necking) before separation.

Cyclic (fatigue) loading of the same material produces an entirely different, largely featureless, low-ductility-looking surface over most of its area: crack growth proceeds incrementally, one small increment of crack advance per cycle at the tip's own local stress-intensity range, governed by the Paris law of Question 5, and leaves the characteristic striations (one per cycle) within a broader field of macroscopic beach marks recording changes in growth rate or load spectrum (Question 6a). No macroscopic necking accompanies fatigue crack growth, because the nominal stress driving it is well below yield; only in the final stage, once the remaining ligament is too small to carry the load, does the fracture transition abruptly to the dimpled overload mode identical to the static case — so a single fatigue fracture surface typically shows both signatures, striated over most of its area and dimpled only in the small final-overload region.

The essential contrast, then, is void nucleation/growth under one rising monotonic load (static overload, fully dimpled, macroscopic necking) versus incremental, striated crack advance under many small load cycles (fatigue, mostly striated with a small final dimpled region, no macroscopic necking) — the same underlying ductile-fracture physics (microvoid coalescence) appears in both, but only in the small overload region of a fatigue surface, never as the dominant mechanism.

7.2 — (b) Derivation: necking begins at $\varepsilon = n$

Approach. Plastic instability (necking) begins at the maximum load the specimen can sustain, i.e. where the load $P=\sigma A$ passes through a stationary point, $dP=0$. Express that condition in terms of true stress and true strain using volume constancy, then substitute the given hardening law.

  1. Instability condition from $dP=0$. With $P=\sigma A$, $dP=A\,d\sigma+\sigma\,dA=0$, so $$\frac{d\sigma}{\sigma}=-\frac{dA}{A}$$
  2. Relate $-dA/A$ to true strain via volume constancy. Constant volume ($AL=\text{const}$) gives $dA/A=-dL/L=-d\varepsilon$ (true strain $d\varepsilon=dL/L$), so $-dA/A=d\varepsilon$, and the instability condition becomes the well-known Considère criterion: $$\frac{d\sigma}{d\varepsilon}=\sigma$$
  3. Substitute the given hardening law $\sigma=K\varepsilon^n$. Differentiating, $$\frac{d\sigma}{d\varepsilon}=Kn\varepsilon^{n-1}$$ Setting this equal to $\sigma=K\varepsilon^n$: $$Kn\varepsilon^{n-1}=K\varepsilon^{n}$$
  4. Cancel $K\varepsilon^{n-1}$ from both sides (valid for $\varepsilon\ne0$). $$n=\boxed{\varepsilon}$$

Necking therefore initiates at a true strain numerically equal to the work-hardening exponent $n$, independent of $K$. This is why $n$ is reported as a formability parameter in its own right: a higher-$n$ material work-hardens fast enough to keep redistributing strain away from an incipient neck for longer, sustaining more uniform elongation before instability, which is exactly why high-$n$ alloys (e.g. annealed austenitic stainless steel, $n\approx0.4$-0.5) are preferred for deep-drawing and other severe sheet-forming operations over low-$n$ alloys of similar strength.