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.
Metal microstructure evolution. Below yield, deformation is elastic lattice strain with no permanent dislocation motion. At yield, dislocations begin to glide on the favoured slip systems; as straining continues to UTS, dislocation density multiplies rapidly (Frank–Read sources), dislocations intersect, tangle and pile up at grain boundaries and each other, and this "forest hardening" is the microstructural origin of the strain-hardening (rising) portion of the curve. Once the strain-hardening rate falls below the flow stress itself (Considère's instability condition, derived in Question 7(a)), deformation localises into a neck; grains within the neck elongate and rotate toward the tensile axis until final ductile fracture (microvoid coalescence) occurs there.
Polymer microstructure evolution. The initial low-modulus elastic region reflects bond bending/stretching and limited chain uncoiling within the amorphous regions between crystalline lamellae. At the yield point, crystalline lamellae begin to slip, tilt and eventually fragment into smaller blocks, while tie-molecules connecting lamellae are pulled taut; this produces the characteristic local stress drop as a "neck" forms and then propagates along the gauge length at roughly constant true stress (cold drawing) — unlike a metal's neck, which localises and worsens rather than propagating. Within the drawn material, the fragmented lamellae and folded chains unfold and align into a fibrillar microstructure oriented along the load axis; once most chains are aligned, further extension loads covalent backbone bonds directly, producing the final strain-hardening upturn to fracture.
(i) Stiffness. Under axial (isostrain) loading, fibre and matrix strain together, and the rule-of-mixtures modulus $E_c=V_fE_f+V_mE_m$ is a strain-weighted average dominated by whichever phase is stiffer. Since carbon fibre's modulus ($E_f\sim230$–400 GPa) is one to two orders of magnitude above a typical polymer matrix ($E_m\sim3$ GPa), even a modest fibre volume fraction raises the composite stiffness far above the matrix alone.
(ii) Strength. Provided the fibres are continuous (or long enough to exceed the critical length for full stress transfer) and well bonded, the stiff fibres carry the great majority of the applied load once strained past the matrix's own low-strain elastic range; the matrix's role is to transfer shear stress into the fibres across the interface and to keep the fibres aligned and buckling-resistant in compression. The resulting composite strength scales with the fibre strength weighted by $V_f$ (minus a small matrix contribution), which is far higher than the strength of the unreinforced matrix.
(iii) Fracture toughness — greater than both constituents. The ceramic fibre alone is intrinsically brittle (very low $K_{Ic}$, near-zero plastic energy absorption) and the polymer matrix alone is comparatively weak. In the composite, however, a growing crack meets a series of extrinsic, energy-absorbing mechanisms unavailable to either constituent in isolation: crack deflection and blunting at the (deliberately weak) fibre-matrix interface, matrix microcracking between fibres, fibre bridging of the crack faces behind the tip, interfacial debonding, and frictional fibre pull-out as bridging fibres eventually fail and slide out. Each of these mechanisms dissipates additional energy per unit crack advance, so the composite's net work of fracture substantially exceeds that of the brittle fibre alone (which offers essentially none of these mechanisms) and also exceeds the unreinforced matrix (which lacks the fibres to bridge and pull out).
