Question 6 of 8: Metal vs. Polymer Stress-Strain Behaviour; CFRP Composite Properties

Question 6: Metal vs. Polymer Stress-Strain Behaviour; CFRP Composite Properties (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) Stress-strain curves and accompanying microstructural change

(i) Tough polycrystalline metallic alloy. Below the elastic limit, deformation is accommodated by reversible lattice (and, at grain boundaries, some reversible boundary) strain; the microstructure is essentially undisturbed. At yield, dislocations already present begin to glide on the most favourably stressed slip systems, and dislocation sources (e.g. Frank–Read sources) begin operating and multiplying the dislocation population. Through the hardening region up to UTS, gliding dislocations increasingly intersect one another (forming jogs and forest obstacles), pile up against grain boundaries and second-phase particles, and the dislocation density rises by orders of magnitude; grains rotate and elongate somewhat in the tensile direction, and in an alloy containing second-phase particles, local strain concentrates around them. At UTS the strain-hardening rate has fallen to equal the current flow stress (Considère's criterion, Question 1(a)), and beyond that point deformation localises into a neck, where void nucleation at inclusions/particles, growth and coalescence proceed rapidly to ductile (dimpled) fracture.

(ii) Semicrystalline polymer. In the elastic region, deformation is small elastic bond-stretch and bond-angle distortion within the amorphous tie-chain regions plus limited stretching of the folded-chain lamellae — a much lower modulus than a metal because it is dominated by weak secondary (van der Waals) bonding between chains rather than primary bonds. At yield, the spherulitic lamellar structure begins to break up: lamellae tilt and slide, and fine crystalline blocks separate from the parent lamellae. A localised neck then forms and, unlike a metal, propagates along the gauge length at a roughly constant engineering stress (the "cold-drawing" plateau) as material ahead of the neck is progressively drawn into it: the folded-chain lamellar structure unfolds and the chains and remaining crystalline blocks reorient (draw) into a fibrillar structure aligned with the tensile axis. Once most of the gauge length has been drawn into this oriented, fibrillar microstructure, further extension requires stretching the now highly aligned covalent chain backbones directly, which produces a final, steep orientation-hardening rise in stress before fracture — the polymer analogue of dislocation forest-hardening, but achieved by molecular alignment rather than dislocation multiplication.

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

(i) Stiffness. The fibre modulus $E_f$ is typically one to two orders of magnitude above the matrix modulus $E_m$. Loaded along the fibre direction under the iso-strain (equal-strain, parallel-spring) assumption, the rule of mixtures gives $E_c=V_fE_f+V_mE_m$, which is dominated by the fibre term even at moderate fibre volume fraction $V_f$: the stiff fibres simply carry almost all of the load at the shared strain, so the composite as a whole deflects far less than the neat resin under the same stress.

(ii) Strength relative to the matrix. Because the fibres are both stiffer and (for a well-chosen reinforcement) far stronger than the matrix, at any given applied strain the fibres are carrying a disproportionate share of the stress; the matrix's role becomes transferring load into the fibres by interfacial shear rather than carrying it. For carbon/epoxy the fibres' failure strain (about 1.5%) is below the resin's, so the longitudinal strength is $\sigma_c\approx V_f\sigma_f^{*}+V_m\sigma_m'$, where $\sigma_m'$ is the matrix stress at the fibre failure strain; with fibre strengths of several GPa against a resin strength of order 50–80 MPa this is many times the strength of the unreinforced polymer. The matrix also protects the fibres from surface damage and, when an individual fibre breaks, redistributes its load to neighbouring fibres over a short shear-transfer length, so isolated fibre breaks do not trigger failure of the whole section.

(iii) Toughness relative to both constituents. This is the least intuitive of the three, because a simple rule-of-mixtures average of two low-toughness ingredients (a brittle ceramic/carbon fibre with $K_{IC}$ of a few MPa$\sqrt{\text{m}}$ or less, and a moderately tough but far weaker polymer) would predict a modest, in-between value — not a value exceeding either. The composite instead gains several extrinsic energy-absorbing mechanisms that do not exist in either constituent alone: crack deflection at the weak fibre–matrix interface (Cook–Gordon mechanism), which blunts and redirects an approaching crack rather than letting it run straight through; fibre bridging, where intact fibres span a matrix crack behind its tip and continue carrying load, shielding the crack tip; and fibre pull-out, where fibres debond and slide frictionally out of the matrix as the crack opens further, dissipating substantial energy through interfacial friction over the pull-out length. None of these mechanisms is available to the ceramic fibre on its own (which simply cleaves) or to the polymer on its own (which has no fibres to bridge or pull out); they are a structural, architecture-dependent property of the composite, which is why fracture toughness in a well-designed fibre composite is superadditive rather than a weighted average of its constituents' own toughness values.