PSI - Issue 2_A

382 Timothy Crump et al. / Procedia Structural Integrity 2 (2016) 381–388 Crump / Structural Integrity Procedia 409 (2016) 000–000 ܩ ሺ ݐ ሻ ൌ ଵ ଶ ௗ ௗ ௔ ቂ గ ఙ మ ா ௔ െ ௞ ଶ ߩ ܽ ଶ ߥ ଶ ቀ ఙ ா ቁቃ ൌ ʹȞ (1) This led on to work by Kanninen (1974) and Freund (1992) via Yoffe (1951) and Cragg (1960) to develop the concept of a limiting crack speed and the first full analytical solution for propagating crack in time, based on the 1D wave solution for a string. The limiting speed for a crack was considered to be a function of the Rayleigh Wave such that: ܽሶ ൌ ܿ ௥ ቀͳ െ ௔ బ ௔ ቁ (2) where ܽ ଴ is the initial crack size and c r is the Rayleigh speed. However, multiple experiments by Kanninen (1974) and more recently Ravi-Chandar (2004) showed that the maximum crack speed tended not to that of the speed of sound within the material but around ~0.7Cr due to multiple mechanisms at microstructure scale (Fig 1(a)), including micro-cracking ahead of a crack-tip caused by high stress levels not allowing time for plastic deformations to occur. This then led on to a set of experiments in Polymethyl methacrylate (PMMA) by Zhou et al (2005) to develop a geometrically independent phenomenological law illustrated in Fig.1.(a). This law was able to capture the velocity hardening ahead of a crack-tip such that: ܩ ௖ ሺܽ ଴ ሶ ሻ ൌ ܩ ଴ ݈݋݃ ቀ ௔ ಽ ሶ ௔ ಽ ሶ ି௔ బ ሶ ቁ (3) where G c is considered a material constant through K I . If the external energy is supplied to a crack propagation, this would require a crack speed beyond this limit, then fragmentation often occurs in the form of macro-crack branching. For a crack in a finite body, which is to say a propagating crack’s influence is felt on its boundaries, this energy could come from reflected waves from the previous displacement of the crack which produces 3 main forms of wave: dilation, shear (including Rayleigh waves) and flexure waves, see Fig.1.(b). Each of these has a different decay rate and intensity. For instance, if one considers a linear elastic material with Possion’s ratio υ = 0.25: the Rayleigh wave at the material surface carries the majority of the energy ~67% with a decay rate of r 0.5 , the bulk shear wave carries 26% with a decay rate of r -2 and the longitudinal pressure wave carries around 7% with a decay rate of r -2 , as seen in Fig.1., Meyers (1994). These are then followed by ‘tertiary’ flexure geometry effects which dissipate the released energy over modal gradients produced by the length of the component. 2

(b)

(a)

Fig. 1. (a) Geometry independent dynamic velocity hardening curve, Zhou (2005); (b) Energy dissipation due to a propagating crack (2D), Meyers (1994)

The majority of approaches to incorporate dynamic crack propagation into global dynamic models have focused on integrating accurately the process at the meso-scale into a globally energetically stable system. This has presented multiple modelling challenges from discrete elements struggling with accurate discontinuity resolution (a propagating crack) and boundary control, to finite element approaches suffering from a high degree of mesh dependency, Song (2008) and Agwai (2011). To overcome these, Zhou et al (2014) suggested accounting for rate