According to the classical theories of elastic or plastic structures made from a material with non-random strength (ft), the nominal strength (σN) of a structure is independent of the structure size (D) when geometrically similar structures are considered.The nominal strength of a structure (σN) has units of stress and is related to the maximum load (Pmax) that the structure can support. For structures that can be approximated as two-dimensional, σN = Pmax/bD where b is the thickness of the two-dimensional structure. For three-dimensional structures, σN = Pmax/D2. Any structure dimension can be chosen for D but it must be homologous for every size. Any deviation from this property is called the size effect. For example, conventional strength of materials predicts that a large beam and a tiny beam will fail at the same stress if they are made of the same material. In the real world, because of size effects, a larger beam will fail at a lower stress than a smaller beam.

The structural size effect concerns structures made of the same material, with the same microstructure. It must be distinguished from the size effect of material inhomogeneities, particularly the Hall-Petch effect, which describes how the material strength increases with decreasing grain size in polycrystalline metals.

The size effect can have two causes: statistical, due to material strength randomness, likelihood of a critical flaw occurring in a high-stress location, and increasing volume increasing the probability of a serious flaw. energetic (and non-statistical), due to energy release when a large crack or a large fracture process zone (FPZ) containing damaged material develops before the maximum load is reached.

Statistical Theory of Size Effect in Brittle Structures

The statistical size effect occurs for a broad class of brittle structures that follow the weakest-link model. This model means that macro-fracture initiation from one material element, or more precisely one representative volume element (RVE), causes the whole structure to fail, like the failure of one link in a chain (Fig. 1a). Since the material strength is random, the strength of the weakest material element in the structure (Fig. 1a) is likely to decrease with increasing structure size (as noted already by Mariotte in 1684).

Denoting the failure probabilities of structure as and of one RVE under stress as , and noting that the survival probability of a chain is the joint probability of survival of all its links, one readily concludes that

The key is the left tail of the distribution of . It was not successfully identified until Weibull in 1939 recognized that the tail is a power law. Denoting the tail exponent as , one can then show that, if the structure is sufficiently larger than one RVE (i.e., if ), the failure probability of a structure as a function of is

Eq. 2 is the cumulative Weibull distribution with scale parameter and shape parameter ; = constant factor depending on the structure geometry, = structure volume; = relative (size-independent) coordinate vectors, = dimensionless stress field (dependent on geometry), scaled so that the maximum stress be 1; = number of spatial dimensions ( = 1, 2 or 3); = material characteristic length representing the effective size of the RVE (typically about 3 inhomogeneity sizes).

The RVE is here defined as the smallest material volume whose failure suffices to make the whole structure fail. From experience, the structure is sufficiently larger than one RVE if the equivalent number of RVEs in the structure is larger than about ; = number of RVEs giving the same if the stress field is homogeneous (always , and usually ). For most normal-scale applications to metals and fine-grained ceramics, except for micrometer scale devices, the size is large enough for the Weibull theory to apply (but not for coarse-grained materials such as concrete).

From Eq. 2 one can show that the mean strength and the coefficient of variation of strength are obtained as follows:

(where is the gamma function) The first equation shows that the size effect on the mean nominal strength is a power function of size , regardless of structure geometry.

Weibull parameter can be experimentally identified by two methods: 1) The values of measured on many identical specimens are used to calculate the coefficient of variation of strength, and the value of then follows by solving Eq. (4); or 2) the values of are measured on geometrically similar specimens of several different sizes and the slope of their linear regression in the plot of versus gives . Method 1 must give the same result for different sizes, and method 2 the same as method 1. If not, the size effect is partly or totally non-Weibullian. Omission of testing for different sizes has often led to incorrect conclusions. Another check is that the histogram of the strengths of many identical specimens must be a straight line when plotted in the Weibull scale. A deviation to the right at high strength range means that is too small and the material quasibrittle.

Energetic Size Effect

The fact that the Weibull size effect is a power law means that it is self-similar, i.e., no characteristic structure size exists, and and material inhomogeneities are negligible compared to . This is the case for fatigue-embrittled metals or fine-grained ceramics except on the micrometer scale. The existence of a finite is a salient feature of the energetic size effect, discovered in 1984. This kind of size effect represents a transition between two power laws and is observed in brittle heterogenous materials, termed quasibrittle. These materials include concrete, fiber composites, rocks, coarse-grained and toughened ceramics, rigid foams, sea ice, dental ceramics, dentine, bone, biological shells, many bio- and bio-inspired materials, masonry, mortar, stiff cohesive soils, grouted soils, consolidated snow, wood, paper, carton, coal, cemented sands, etc. On the micro- or nano scale, all the brittle materials become quasibrittle, and thus must exhibit the energetic size effect.

A pronounced energetic size effect occurs in shear, torsional and punching failures of reinforced concrete, in pullout of anchors from concrete, in compression failure of slender reinforced concrete columns and prestressed concrete beams, in compression and tensile failures of fiber-polymer composites and sandwich structures, and in the failures of all the aforementioned quasibrittle materials. One may distinguish two basic types of this size effect.

Type 1: Structures that fail at crack initiation

When the macro-crack initiates from one RVE whose size is not negligible compared to the structure size, the deterministic size effect dominates over the statistical size effect. What causes the size effect is a stress redistribution in the structure (Fig. 2c) due to damage in the initiating RVE, which is typically located at fracture surface.

A simple intuitive justification of this size effect may be given by considering the flexural failure of an unnotched simply supported beam under a concentrated load at midspan (Fig. 2d). Due to material heterogeneity, what decides the maximum load is not the elastically calculated stress at the tensile face, where = bending moment, = beam depth, and = beam width. Rather, what decides is the stress value roughly at distance from the tensile face, which is at the middle of FPZ (2c). Noting that = , where = stress gradient = and = intrinsic tensile strength of the material, and considering the failure condition = , one gets = where , which is a constant because for geometrically similar beams = constant. This expression is valid only for small enough , and so (according to the first two terms of the binomial expansion) one may approximate it as

which is the law of Type 1 deterministic size effect (Fig. 2a). The purpose of the approximation made is: (a) to prevent from becoming negative for very small , for which the foregoing argument does not apply; and (b) to satisfy the asymptotic condition that the deterministic size effect must vanish for . Here = positive empirical constant; the values = or 2 have been used for concrete, while is optimum according to the existing test data from the literature (Fig. 2d).

A fundamental derivation of Eq. 5 for a general structural geometry has been given by applying dimensional analysis and asymptotic matching to the limit case of energy release when the initial macro-crack length tends to zero. For general structures, the following effective size may be substituted in Eq. (5):

where = strain gradient at the maximum strain point located at the surface, in the direction normal to the surface.

Eq. 5 cannot apply for large sizes because it approaches for a horizontal asymptote. For large sizes, must approach the Weibull statistical size effect, Eq. 3. This condition is satisfied by the generalized energetic-statistical size effect law:

where are empirical constants (). The deterministic formula (5) is recovered as the limit case for . (Fig. 2d) shows a comparison of the last formula with the test results for many different concretes, plotted as dimensionless strength versus dimensionless structure size .

The probabilistic theory of Type 1 size effect can be derived from fracture nano-mechanics. Kramer's transition rate theory shows that, on the nano-scale, the far-left tail of the probability distribution of nano-scale strength is a power law of the type . Analysis of the multiscale transition to the material macro-scale then shows that the RVE strength distribution is Gaussian but with a Weibull (or power-law) left tail whose exponent is much larger than 2 and is grafted roughly at the probability of about 0.001.

For structures with , which are common for quasibrittle materials, the Weibull theory does not apply. But the underlying weakest-link model, expressed by Eq. (1) for , does, albeit with a finite , which is a crucial point. The finiteness of the weakest-link chain model causes major deviations from the Weibull distribution. As the structure size, measured by , increases, the grafting point of the Weibullian left part moves to the right until, at about , the entire distribution becomes Weibullian. The mean strength can be computed from this distribution and, as it turns out, its plot is identical with the plot of Eq. 5 seen in Fig. 2g. The point of deviation from the Weibull asymptote is determined by the location of the grafting point on the strength distribution of one RVE (Fig. 2g). Note that the finiteness of the chain in the weakest-link model captures the deterministic part of size effect.

This theory has also been extended to the size effect on the Evans and Paris' laws of crack growth in quasibrittle materials, and to the size effect on the static and fatigue lifetimes. It appeared that the size effect on the lifetime is much stronger than it is on the short-time strength (tail exponent is an order-of-magnitude smaller).

Type 2: Structures in which a large crack or notch exists

The strongest possible size effect occurs for specimens with similar deep notches (Fig. 4b), or for structures in which a large crack, similar for different sizes, forms stably before the maximum load is reached. Because the location of fracture initiation is predetermined to occur at the crack tip and thus cannot sample the random strengths of different RVEs, the statistical contribution to the mean size effect is negligible. Such behavior is typical of reinforced concrete, damaged fiber-reinforced polymers and some compressed unreinforced structures.

The energetic size effect may be intuitively explained by considering the panel in Fig. 1c,d, initially under a uniform stress equal to . Introduction of a crack of length , with a damage zone