# Size Effect for Failures Characterized by Plasticity, Strength, or Allowable Stress

Consider a family of geometrically similar structures subjected to proportional loading characterized by the nominal stress а/у. Assume that the response of the material can be fully described by a certain constitutive equation relating the stress and strain tensors. No restrictions other than rate independence are imposed on this relation. It may be linear or nonlinear elastic, clasto-plastic, or of some other more sophisticated kind. The point is that the constitutive equation and fracture criterion contain parameters which either are dimensionless (hardening exponents, strain thresholds, etc.) or have the dimension of stress (elastic moduli, yield stresses), but do not contain any parameter with dimension of length. In other words, no characteristic size exists.

Consider a reference structure of size D and a scaled geometrically similar structure of size D’ — XD. Assume that for the reference structure (that of size D) the stress at an arbitrary point of coordinates (хі, хг) for a given load characterized by ct, v is given by aij(<JN, X,x2). This stress distribution satisfies the equilibrium equations and the traction-imposed boundary conditions. Considering the scaled structure, it turns out that if the stresses at homologous points of coordinates x — A. iq and x2 = Xx2 is taken to be identical to those for the reference structure, then this state corresponds to the actual solution for the second structure. This correspondence may be analytically stated as:

<Ty (crff, x ,x2) — x, xf) with x = Ax’i and x2 — x2 (2.3.2)

Then, it is easy to prove that, with this condition, (1) the traction-imposed boundary conditions are auto­matically satisfied because of the similitude (the imposed surface tractions at the boundary are identical at homologous points); (2) the equilibrium conditions <jy, y = 0 are also trivially verified; (3) the constitutive equation is also satisfied if the strdin fields are related by an equation similar to (2.3.2):

£у(<7дг, x’,,Xj) = £ij(oк, x,x2) with x’, = Axi and x2 = Xx2 (2.3.3)

where Єі2 and are the strain tensors for the structures of sizes D and D’, respectively; and (4) the last equation is verified if the similitude law for the displacements is given by

Therefore, the laws of similitude (2.3.2) and (2.3.3) just state that for a given nominal stress <tn, the stresses and strains at homologous points of two geometrically similar structures are identical. This implies, in particular, that the stress and strain maxima and minima also occur at homologous points.

If failure is assumed to occur when the stress, strain, or, in general, a certain function of the two Ф(<ту, £y) reaches a critical value, i. e., when:

Ф(<ту, гу) = Фс (2.3.6)

where Фс is a given critical or allowable value, then the two similar structures will fail at the same nominal stress. Thus, for theories such as plasticity or elasticity with strength limit or allowable stress, the nominal strength of two similar structures of sizes D and D’ is identical:

t’v» = ONv. (2.3.7)

In such a case we say that there is no size effect.

The foregoing result may be also obtained from dimensional analysis. Indeed in this kind of problem, the external load at fracture is completely determined by cr^u, D and a number of geometrical ratios 7; (those defining the shape, for example, the span-to-depth ratio). The material response is determined by a certain critical or allowable stress <rc and a number of dimensionless ratios p. i (the hardening exponent, strain parameters, and ratios of elastic modulus to the allowable stress, E/oc, for example). With these variables, the only dimensionally correct expression for the nominal strength is

VNu = <ГсФ{іі, іч) (2.3.8)

where ф(7j, /і*) is a dimensionless function. Since the arguments 7are geometrical ratios that remain constant for geometrically similar structures, and since the yzj are constants for a given material, (2.3.7) follows.