Shear-lag Theory Research Articles

A fibre composite model of drawn crystalline polymers

The two-phase fibre composite model proposed by us elsewhere [Barham and Arridge (1977)] is used to explain the behaviour of polyethylene and polypropylene on drawing to very high draw ratios. The basis of the theory is that, after necking, there exists a uniform distribution of aligned needle-like elements of near perfect crystallinity in a less-perfect matrix. On further drawing these reinforcing elements are supposed to elongate in a homogeneous deformation while contracting laterally. The resultant change in their aspect ratio is sufficient, using short fibre reinforcement theory, to account for the increase in the modulus on drawing. The Young's modulus of the drawn fibre is given by the expression: E f=cE c 1− tanh x x +(1−C)E m where c is the concentration of elements, E c the modulus of the crystalline elements and E m that of the matrix while x, derived from shear-lag theory, depends upon the aspect ratio of the elements and the moduli of the two phases. For both polyethylene and polypropylene it is found that x ∝ t 3 2 , where t is the post-neck draw ratio. The model is applied (1) to the explanation of the observed relation between Young's modulus and draw ratio, (2) to explain the temperature dependence of drawing behaviour, (3) to postulate a mechanism far non-linear viscoelasticity and creep behaviour and (4) to explain the self-stiffening after annealing under constraint [Arridge, Barham and Keller (1977)].

Deformation behavior of an S‐B‐S copolymer

AbstractIn accordance with previous studies, ‘single crystal’ tyr samples were prepared from the polystyrene‐polybutadien‐ polystyrene (S‐B‐S) three block copolymer Kraton 102 consist ing of a hexagonally packed parallel array of glassy S cylinder within a rubbery B matrix. Following earlier investigations o the pronounced anisotropy of the modulus, the actual deformation mechanism is examined in terms of the microstructures on straining perpendicular and parallel to the cylinder direction up to and beyond yielding. On perpendicular straining, a one to‐one relation is observed between ‘lattice’ and sample stray up to about 10 percent, coupled with complete recovery. For higher strains, the deformation becomes increasingly irreversible dependent on the time of stressing, a behavior which is found to be caused by cracks forming along crys tallographically defined directions of the hexagonal macrolai tice. On parallel straining, the stress‐strain curves display yield behavior at ∼3 percent strain, the yielded material be coming more compliant even at small strains on repeated test ing. However, on storage, the sample rehardens, a proces accelerated by heat treatment. This behavior is explained in terms of a breaking up and reformation of the cylinders and is supported by the subsequent experiments. These includ measurement of birefringence, which in the reversible small strain region corresponds to expectations from uniform strain in both S and B phases, but which beyond the yielded region displays behavior such as expected from deformation of the B phase alone. The postulated break‐up of cylinders on yielding is directly confirmed by electron microscopy on Iongitudinal sections thin enough to contain a single layer of cylinders. The yield behavior itself is treated theoretically using the composite mechanics approach. Two treatments are applied, the trad itional shear lag theory and a new random rod breaking theory In the latter case, in particular, very good agreement with experiment is found in that both the observed stress to yield and the electronmicroscopically seen fractured rod length in the yielded product are predicted.

On the stress concentration factor in fibrous composites

Abstract In order to presume a probabilistic strength of a composite material, it is necessary to know the stress states in a composite plate. In the present paper, the stress states in a model, unidirectionally reinforced with continuous fibres, except for some discontinuous fibres, and subjected to a tensile load parallel to the fibre direction, are considered. A stress concentration factor and an ineffective length in the model are calculated using a new method proposed previously by the authors. Comparison is made with experimental values and results by means of shear-lag theory.

The measurement of shear modulus in highly anisotropic materials: the validity of St. Venant's principle

Torsion pendulum measurements of the longitudinal shear modulus in 'single-crystal' samples of an S-B-S block copolymer are reported. These samples consisted of a hexagonal array of polystyrene cylinders embedded in a matrix of polybutadiene, with the cylinder axes uniaxially aligned. Sample morphology and associated anisotropic physical properties resemble those of a miniature fibre-reinforced system. The shear modulus calculated from the torsional stiffness by conventional means, assuming St Venant's principle, is found to depend upon the sample length/width ratio, the results for short specimens being double those for very long specimens. It appears that St Venant's principle expressed in its usual form is not applicable to materials possessing a large mechanical anisotropy. An alternative interpretation of experimental results is given by invoking the shear-lag theory of fibre reinforcement.

Axial creep and stress-rupture of B-Al composites

Constant load creep and stress-rupture properties of boron and silicon carbide coated boron, BORSIC®, filament reinforced aluminum alloys were measured at 300°, 400°, and 500°C. The mean stresses for 1000 hr rupture life at the three test temperatures were 96,000, 82,000, and 72,000 psi, respectively. Elongation in creep tests was checked by using scribed lines, and this technique indicated that grip-mounted extensometers were inadequate for precision measurement due to shear of aluminum in the grips. A maximum of 0.2 pct plastic elongation was measured, which corresponds to the creep properties of the filaments. Comparison of Borsic composite results with data for composites fabricated with uncoated boron filament showed that the short-time rupture strengths of the two materials were similar, but that the stress for rupture of the boron filament composites decreased much more rapidly with increasing rupture time than did the stress for rupture in the Borsic filament composites, due to degrading reactions of the uncoated boron with the matrix. Fracture analysis was performed using optical metallography and scanning electron microscopy. The analysis indicated that the major portion of filament failures in the region of the fracture surface at elevated temperatures initiated at the outer surface of the filament, whereas filament failures that initiated in the region of the core were more common in room temperature tensile specimens. The amount of plastic flow in the matrix increased markedly with increasing temperature, but filament pullout lengths were far less than those predicted by a simple shear-lag theory on the basis of the shear strength of the matrix.