Question 6 of 8: Metal versus Semicrystalline Polymer; Why Fibre Composites Win

Question 6: Metal versus Semicrystalline Polymer; Why Fibre Composites Win (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.

6.1 — (a) The two curves on one set of axes, and the microstructure behind each

The metal, from the elastic limit to the UTS. Below $\sigma_y$ the response is the reversible stretching of interatomic bonds, and the slope is $E$ — a genuine material constant of order 70–210 GPa, insensitive to microstructure. At the elastic limit, dislocation sources begin to operate in the grains most favourably oriented for slip, and yield propagates from grain to grain as pile-ups raise the stress in the neighbours; in a low-carbon steel this is the discontinuous Lüders behaviour, in most other alloys a smooth transition. Beyond yield the curve rises — work hardening — because the dislocations generated by the deformation get in one another's way. The dislocation density climbs from perhaps $10^{10}$ to $10^{15}$ m${}^{-2}$; forest dislocations cut through the moving ones and leave jogs; the moving dislocations organise into tangles and then into a cell structure whose walls are dense and whose interiors are relatively clean, and the cell size falls as strain increases. The flow stress follows the Taylor relation $\tau = \tau_0 + \alpha G b\sqrt{\rho}$, so it rises as the square root of dislocation density. Meanwhile the grains themselves elongate along the tensile axis and develop a crystallographic texture.

The UTS is reached when the geometric loss of cross-sectional area exactly cancels the material's ability to harden — the Considère condition $\mathrm{d}\sigma/\mathrm{d}\varepsilon = \sigma$ in true-stress terms. Beyond it deformation localises into a neck; the triaxial stress state inside the neck nucleates voids at inclusions and second-phase particles, the voids grow and coalesce along the neck's centreline, and the final separation is the shear lip that gives the classic cup-and-cone fracture. Importantly, the material inside the neck is still hardening — the fall in the nominal curve after the UTS is entirely a geometric artefact of dividing by the original area.

The semicrystalline polymer, from the elastic limit to the UTS. The starting microstructure is completely different: chain-folded crystalline lamellae 10–20 nm thick, arranged radially into spherulites tens of micrometres across, separated by amorphous layers containing entanglements and tie molecules. Below yield, the deformation is the stretching of the amorphous layers and of the lamellae themselves, and the modulus — 0.2–2 GPa, a hundredth of the metal — is dominated by the compliant amorphous phase and is strongly time- and temperature-dependent, because it is viscoelastic rather than purely elastic.

Yield in the polymer is a genuine maximum, not a plateau: the curve rises to a peak and then drops. Microstructurally, at the peak the amorphous tie chains have been pulled taut, the lamellae begin to tilt, slip and separate, and blocks of lamellae break away and start to align with the tensile axis. This happens first at the weakest point, so a neck forms. Then comes the behaviour that has no metallic analogue: instead of running away, the neck stabilises and propagates. Inside the drawn region the spherulitic structure has been destroyed and replaced by a fibrillar morphology — crystal blocks strung along highly oriented tie chains, all parallel to the draw direction — which is far stronger in that direction than the undrawn material. The neck therefore cannot deepen; it can only spread, consuming undrawn material at both shoulders at an almost constant load, which is the flat plateau on the curve. The natural draw ratio, typically 4–8, fixes the plateau strain. Only when the whole gauge length has been drawn does the stress rise again — strain hardening by the now fully aligned chains — and the true ultimate tensile stress is reached just before fracture, at a strain that may exceed 400 per cent.

The comparison in one sentence. The metal is strong, stiff and modestly ductile, and hardens by multiplying line defects; the polymer is compliant and weak but enormously extensible, and hardens by converting an isotropic spherulitic texture into an oriented fibrillar one. Both absorb a comparable amount of energy per unit volume, for very different reasons — the metal by a large stress over a small strain, the polymer by a small stress over a large strain.

6.2 — (b) Why a ceramic-fibre-reinforced plastic beats both of its constituents

Given. A representative unidirectional CFRP for the illustration below: fibre volume fraction $V_f = 0.60$, fibre modulus $E_f = 230$ GPa with tensile strength $\sigma_f^{*} = 3500$ MPa, epoxy matrix modulus $E_m = 3.5$ GPa with tensile strength 70 MPa.

Find. The axial modulus and strength predicted by the rule of mixtures, and the microstructural reason why the composite's fracture toughness exceeds that of either phase.

(i) Stiffness. With the fibres continuous and aligned with the load, both phases are constrained to the same strain (an isostrain, or Voigt, arrangement), so the loads add in proportion to area:

which is 39.8 times the modulus of the matrix alone and comparable with steel at one-fifth of the density. The mechanism is simply that a stiff phase in parallel with a compliant one carries essentially all of the load: at equal strain the fibres here carry $138.0/139.4 = 99$ per cent of it. Nothing subtle is happening — the composite is stiff because the ceramic fibre is stiff, and the polymer's only job is to hold the fibres in place. (Loaded transversely the arrangement becomes isostress, the reciprocal rule applies, and the modulus collapses towards $E_m$; this anisotropy is why real structures are laminated.)

(ii) Strength. The same isostrain argument sets the strength, with one refinement: because the fibres are brittle and the matrix is not, the fibres fail first, at a strain $\varepsilon_f^{*} = \sigma_f^{*}/E_f = 3500/230000 = 1.52$ per cent. The matrix is only carrying $E_m\varepsilon_f^{*} = 53.3$ MPa at that moment, well below its own 70 MPa strength, so

about 30 times the matrix strength. The deeper reason a fibre is strong at all is a fracture-mechanics one: the strength of a brittle solid is set by its largest flaw through $\sigma \approx K_{Ic}/\sqrt{\pi a}$, and drawing the ceramic into a filament a few micrometres in diameter physically caps the flaw size at the fibre radius. A bulk rod of the same glass or graphite contains flaws a thousand times larger and is a thousandth as strong. The composite therefore captures the intrinsic strength of the ceramic bond, which the bulk ceramic cannot.

(iii) Fracture toughness. This is the part that is not a rule of mixtures, and the examiner's phrasing — “relative to both phases” — is the clue. Bulk ceramic has $G_{Ic}\approx 10$–$50$ J m${}^{-2}$; a cast epoxy has perhaps 100–300 J m${}^{-2}$; a unidirectional CFRP loaded across the fibres has 10–100 kilojoules per square metre. The excess comes from a set of energy-absorbing events that only exist because the two phases are present together, and all of them happen at the interface:

• Crack deflection and interfacial debonding. A crack running through the matrix towards a fibre meets a weak interface. The tensile stress ahead of a crack tip acts normal to the interface, so the interface opens before the crack arrives (the Cook–Gordon mechanism) and the crack is turned through 90° to run harmlessly along the fibre. A single straight crack path is replaced by a long, tortuous one.
• Fibre bridging. Fibres that have not yet failed span the crack faces behind the tip and carry load across the opening, reducing the stress intensity actually experienced at the tip.
• Fibre pull-out. When a bridging fibre finally breaks, it breaks at its own weakest flaw, generally not in the crack plane, so the broken end must then be dragged out of the matrix against interfacial friction $\tau_i$. The work done is large: for fibres shorter than the critical length $\ell_c = \sigma_f^{*}d/2\tau_i$, the pull-out contribution to $G_c$ scales as $V_f\tau_i\ell^{2}/d$, and pull-out alone typically accounts for most of the composite's toughness.
• Matrix plasticity and, in a laminate, delamination spread the damage over a large volume instead of concentrating it at one tip.

The essential and slightly counter-intuitive point is that the interface must be deliberately weakened — through the choice of fibre sizing — to obtain toughness. Bond the fibres perfectly to the matrix and the crack sails straight through them; the composite then has the strength predicted above and the toughness of a monolithic ceramic. Toughness is bought with controlled interfacial weakness, and that is a design decision made in the sizing chemistry, not a property of either constituent.