PSI - Issue 41

Fabrizio Greco et al. / Procedia Structural Integrity 41 (2022) 576–588 Author name / Structural Integrity Procedia 00 (2019) 000–000

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which DSIFs need to be evaluated. In contrast, the second is an auxiliary state with known SIFs. In dynamic fracture mechanics, the auxiliary state can be expressed using a twofold set of analytic solutions: the first set consists of the Williams’ asymptotic solution (Williams (1956)) associated with a straight crack in an infinite plate under remote quasi-static loadings. The second is the analytical solution relative to an advancing crack in an infinitive medium at constant velocity derived by Rise (Rice (1979)). According to Chen et al. (Chen et al. (2019)), the Equivalent Domain Integral (EDI) integral form of the M integral assumes the following expression:         ,1 ,1 1 , ,1 ,1 ,1 ,1 act aux aux act act aux act aux ij j ij j ij ij j j i i A act aux aux act act aux aux act j j j j j j j j A M u u u u q dA u u u u u u u u qdA                                    (12) Using the relationship between the J -integral and DSIFs for two-dimensional crack problems, one obtains:

2

 

 

act II s M K K f a K K f a E          aux act aux I I d II

(13)

where, E E   for plane stress and d s f a f a   are the universal functions, depending on the crack tip velocity a  , whose expressions are reported in Ref. (Yan et al. (2021)). The DSIFs for the actual state ( i.e. , , act act I II K K ) are evaluated by means of two interaction integrals, defined using as auxiliary fields the analytical solutions associated with a pure mode-I and a pure mode-II conditions, thus gaining: 2 (1 ) E E     for plane strain. In addition, ( ), ( )

E M 

E M 

, act aux I 

, act aux II 



 0 ;





0

act

aux I 

act

aux II 

(14)

K

K

K

K









  f a K  d

  f a K 

I

II

II

I

aux I 

aux II 

2

2

I

s

II

3 Numerical implementation The proposed modeling approach has been implemented in Comsol Multi-physics (COMSOL (2018)). This commercially available software offers an easy-to-use FE environment to investigate numerically different structural problems (see, for instance, Greco et al. (2013), Lonetti and Pascuzzo (2014), Bruno et al. (2016), Lonetti et al. (2016), Lonetti and Pascuzzo (2020)). Further, standard Comsol’s functionalities can be easily enhanced through additional apps linking the FE environment with specialist software, such as those dedicated to technical drawing or mathematic analysis. Among these, the LiveLink for MatLab app enables linking Comsol with MatLab, thus permitting to arrange sequences of repetitive Comsol commands within automatized MatLab scripts. This functionality has been adopted in the proposed model to create a script code managing the several steps involved in the propagation process configured by the proposed method. The remaining part of this section describes the steps involved in the crack propagation phase configured by the proposed method. Fig. 3 illustrates sketches about some of the leading steps involved to support the discussion. At first, the script code performs some geometric operations on the geometry inserted in the numerical model. Note that the geometry of the computational domain can be configured either by using the drawing tools available in Comsol or by importing external drawing files created with specialist drawing software. In both cases, the geometry must present an internal pre-crack modeled by a polyline (see Fig. 3-a). The code starts working by splitting the polyline into two entities by inserting an additional node close to the crack tip. This operation produces a short segment that can be stretched during the propagation process, thus reproducing the advancement of the inner material defect (Fig. 3-b). In contrast, the remaining portion of the polyline keeps fixed, thus ensuring that the initial pre-crack surface preserves its initial configuration during crack propagation phase. Next, the code meshes the changed geometry, sets the boundary conditions, and starts the analysis. The Dynamic Stress Intensity Factors (DSIFs) are evaluated for each time step using the M -integral.