numerically efficient modified runge–kutta solver for fatigue crack growth analysis

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Numerically efcient modied Runge–Kutta solver for fatigue crack growth analysis Christian Amann a,, Kai Kadau b a Siemen s AG Energy, 45473 Mülheim, Germany b Siemen s Energy Inc., 5101 Westingho use Boulevard, Charlotte, NC 28273, USA a r t i c l e i n f o  Article history: Received 25 September 2013 Received in revised form 23 January 2015 Accepted 14 March 2016 Available online 15 April 2016 Keywords: Fatigue crack growth Conservative Runge–Kutta Overestimate crack size Probabilistic Fracture mechanics a b s t r a c t We present a modied Runge– Kutta algorithm which yields a conservative estimate of the cra ck siz e for fatigu e crack growth ev en for large integr atio n ste p sizes. Co ns erv ati ve in this con text means to over estimate the crack size. Commo nly used algorith ms (e.g. Euler, Runge–Kutta) usually underestimate the crack growth and only converge for small step siz es to an accurate val ue. As the pre sen ted alg ori thmoveres timate s th e cra ck gro wt h even for large and converges for small integration step sizes, it can be used for instance in Monte-Carlo based probabilistic fracture mechanics simulations, which might be other- wise computational impractical.  2016 Elsevier Ltd. All rights reserved. 1. Introduction In ma ny constructional ap pli cat ions fat igu e crack gro wt h is th e life time limitin g mechanism of a component. The lif eti me of such components in many cases is estimated by the Mode I crack growth in cuboidal geometries  [2,3,5,7,9] . An example geo metry is given in Fig. 1. Mod e I crack gro wt h rat es  d a dN  are us ua lly ch aracte riz ed as a fun cti on f  of th e stres s intensit y ran ge DK . Assuming a power law for  f , the well known Paris law (1) [6]  is obtained. da dN ¼  f ðDK Þ ¼ C  DK m ð1Þ Here,  a  is the crack length,  N  is the number of cycles,  DK  is the stress intensity range and  C  and m are the Paris parameters. The stress intensity range can be expressed as: DK  ¼ Dr  ffiffiffiffiffiffi pa p  Y ðaÞ ð2Þ with  Dr as stress range, and  Y ðaÞ  as a geometry factor. More complex expressions (e.g. Forman– Mettu equation [2] ) or even measured inter polated data points can also be used to describe the experimental measured crack growth rates. In general it is not possible to nd analytical solutions of  (1) for the crack size. Hence solutions are obtained by numeric integration schemes such as Euler, Runge–Kutta 2nd, 4th or higher order algorithms  [1,4,10]. It can been shown that for http://dx.doi.org/10.1016/j.engfracmech.2016.03.021 0013-7944/  2016 Elsevier Ltd. All rights reserved. Corresponding author. E-mail addresses: Christian.Amann@siemens.com (C. Amann),  Kai.Kadau@ siemens.com (K. Kadau). Engineering Fracture Mechanics 161 (2016) 55–62 Contents lists available at  ScienceDirect Engineering Fracture Mechanics journal homepage:  www.elsevier.com/locate/engfracmech

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