doi: 10.1111/j.1460-2695.2010.01475.x Building models for crack propagation under fatigue loads: application to macrocrack growth C. CASTILLO 1 , A. FERN ´ ANDEZ-CANTELI 2 , E. CASTILLO 3 and H. PINTO 3 1 Department of Civil Engineering, University of Castilla-La Mancha, Spain, 2 Department of Construction and Fabrication Engineering, University of Oviedo, Spain, 3 Department of Applied Mathematics and Computational Sciences, University of Cantabria, Spain Received in final form 6 April 2010 ABSTRACT This paper deals with the problem of building crack growth models. First, some incon- veniences of existing models are described. Next, a general methodology is presented starting by identifying the set of variables involved in the crack growth problem obtain- ing a minimum subset of dimensionless parameters with the help of the Buckingham  theorem, and imposing some consistency and compatibility conditions in terms of func- tional equations. These functional equations, once solved, provide the subset of crack growth models satisfying the required deterministic and stochastic compatibility condi- tions, which, in addition to providing mean values of the crack sizes as a function of time, as alternative models do, also give densities of the crack sizes. The main elements required to build a crack growth model, such as the initial crack size distribution, the crack growth function and a loading effect function, have been identified. The methodology is illus- trated with some examples, including crack growth for different load histories. Finally, some models proposed in the past are shown to satisfy these conditions and one numerical example is given. Keywords compatibility conditions; dimensional analysis; functional equations; load history; stochastic crack growth models. INTRODUCTION AND MOTIVATION So far, most crack growth formulas have represented dif- ferent versions of the Paris law. Unfortunately, many of them present dimensional inconsistencies. In some cases this implies serious inconveniences, because the dimen- sions of the parameters are dependent on the values of other parameters, as with the classical Paris law da d N = C K m , (1) where in this case, da dN is the crack growth rate, K is the stress intensity factor range, C is a dimensional parame- ter the dimensions of which surprisingly depend on the value of parameter m. In other words, the dimensions of parameter C cannot be known before parameter m is esti- mated. More precisely, the left-hand side of (1) has length dimension, thus, the right-hand side must have length di- mension too, but because K is not dimensionless, the Correspondence: C. Castillo. E-mail: castie@unican.es dimensions of C cannot be known before the value of the parameter m is known. The same problem occurs with the models proposed in 1–4 This creates certain incon- veniences to users, causes mistakes, and leads to serious doubts as to their physical validity. To avoid these problems, models involving dimension- less parameters 5 should be used. This is what the Buck- ingham theorem, 6–8 which is not only mathematically but physically based, recommends. In summary, the problem with the Paris law is not only that the proportionality con- stant C is not dimensionless, but much more important that its dimensions cannot be known until the exponent parameter m is known (estimated). This means that its dimensions (not its values) are different for all materials because they are dependent on the value of the exponent parameter m, that has no physical sense. Thus, the problem does not lie on the K dimensions, but on the Paris’ law itself, which contains K and then it has to respect the dimensions of K and any other magnitude in it in order the right- and left-hand sides of the Paris law equation to have the same dimensions. c  2010 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 33, 619–632 619 Fatigue & Fracture of Engineering Materials & Structures