Size Effect on Structural Failure

The effect of structure size on its nominal strength is of paramount importance in extrapolating small-scale laboratory tests to full-scale structures. Traditional failure criteria, such as the maximum stress, maximum strain, and strain energy criteria ignore the size effect, in other words, the nominal strength of a structure is independent of the characteristic size of the structure. Such theories failed to explain structural failures that occurred at stress levels considerably lower than the ultimate strength of the material. The question of size effect was first discussed by Leonardo da Vinci (1452–1519) who run tests to determine the strength of iron wires [1]. He found an inverse relationship between the nominal strength and the length of the wire for wires of constant diameter. We quote from an authoritative translation of Leonardo’s sketch book: ‘Observe what the weight was that broke the wire, and in what part the wire broke…Then shorten the wire, at first by half, and see how much more weight it supports; and then make it one quarter of its original length, and so on, making various lengths and noting the weight that breaks each one and the place in which it breaks’. This is of course a strong exaggeration of the actual size effect. The rule of Leonardo was rejected by Galileo in his famous book [2], in which he founded mechanics of materials. Galileo argued that cutting a long cord at various points should not make the remaining part stronger. He pointed out, however, that a size effect is manifested in the fact that large animals have relatively bulkier bones than small ones. Half a century later, Mariotte [3] based on extensive experiments on ropes, paper and tin made the observation that ‘a long rope and a short rope always support the same weight unless that in a long rope there may happen to be some faulty place in which it will break sooner than in a shorter’. He thus initiated the statistical theory of size effect, two and a half centuries before Weibull. The above studies concern with the structural size effect which differs from the material size effect, which was first systematically studied by Griffith [4], who with his key ideas about the strength of solids laid down the foundation of the present theory of fracture. In experiments performed on cracked circular tubes made of glass, Griffith observed that the maximum tensile stress in the tube was of the magnitude of 344 kip in−2 (2372 MPa), while the tensile strength of glass was 24.9 kip in−2 (172 MPa). These results led him to raise the following questions (we quote from reference [4]) ‘If the strength of this glass, as ordinarily interpreted, is not a constant, on what does it depend? What is the greatest possible strength, and can this strength be made for technical purpose by appropriate treatment of the material?’ From Griffith’s experiments it is obtained that the strength increases as the fiber diameter decreases. The maximum strength of glass was found to be 1600 kip in−2 (11 000 MPa), which is two orders of magnitude higher than the ordinary strength of glass of 24.9 kip in−2 (172 MPa). In an effort to explain these phenomena Griffith laid down the principles of a new discipline of engineering design based on fracture mechanics [5]. Fracture mechanics is based on the realistic assumption that all material contain crack-like defects which constitute the nuclei of fracture initiation. A major objective of fracture mechanics is to study the load-carrying capacity of structures in the presence of initial defects, where a dominant crack is assumed to exist. Weibull [6] put the basic framework of the statistical theory of size effect. The theory applies to structures that fail at the initiation of macroscopic fracture and have at fracture only a small fracture process zone causing negligible stress redistribution. This is the case of brittle materials. It does not apply to quasibrittle materials, like concrete and mortar made with various cement and admixtures, polymers, rock, ice, fiber or particulate composites, fiber-reinforced concretes, toughened ceramics, bone, biological shells, stiff clays, cemented sands, grouted soils, coal, paper, wood, wood particle board, various refractories, some special tough metals alloys, filled elastomers, etc, which are characterised by the existence of a large fracture process zone with distributed cracking damage. Weibull’s theory attributing the observed size effect to the randomness of material strength dominated until the early 1980s, when Bažant [7, 8] revisited the size effect of structural strength. He observed that classical theories of solid mechanics including elasticity and plasticity which do not account for size effect do not apply to materials that instead of plastic yielding exhibit softening damage, such as a distributed cracking. In that case, a strong, non-statistical size effect may be caused by stress redistribution creating energy release from an elastically unloaded region of material at the flanks of a propagating damage or large cohesive fracture, taking place before the maximum load is reached. When this deterministic size effect occurs, it normally prevails the size effect due to the statistical distribution of strength, described by the Weibull theory. In order to extrapolate to large sizes from material strength values obtained from laboratory tests of relatively small specimens the size effect should be well understood. Bažant laid down the foundation of a new revolutionary concept with his discovery of a deterministic scaling law based on the release of stored energy due to stable growth of large fractures or large damage zones prior to failure. He derived a simple formula for the size effect law which describes the size effect on nominal strength of structures made of quasibrittle materials, which are characterised by the existence of a sizable fracture process zone at the tip of the macroscopic crack. Quasibrittle materials are brittle materials characterised by a fracture process zone (FPZ) that is not negligible compared to the characteristic size D (or cross-sectional dimension) of the structure. Typically, the size of the FPZ, taken as the material characteristic length lch, is about 5 to 50 times the maximum inhomogeneity size, and quasibrittle behavior is observed only for D/lch ∼ 1 to 1000. For larger D/lch, the FPZ can be regarded as a point and then the behavior is brittle, while for D/lch < 1, and approximately for up to about 5, the behavior can be regarded as quasi-plastic. Bažant found that in such materials the size effect is transitional between plasticity (for which there is no size effect) and linear elastic fracture mechanics (for which the size effect is the strongest). The curve of the logarithm of the nominal strength versus the logarithm of the size represents a smooth transition from a horizontal asymptote corresponding to the strength criterion (plastic limit analysis) to an inclined asymptote of −0.5 slope, corresponding to linear elastic fracture mechanics. Bažant studied the size effect on the tensile [9] and flexural [10] strength of fiber-composite composite laminates, on compression strength of fiber composites failing by kink band propagation [11], on fracture of closed-cell polymeric foams [12], on compressive strength of sandwich panels [13], and on cohesive delamination triggered by sandwich skin wrinkling [14]. For a recent review of the work performed on the scaling and size effect in the failure of composites, foams and sandwich structures, the reader is referred to [15].