On the challenging FEM application fields in the fracture mechanics

Vladislav Kozák
Institute of Physics of Materials,  Academy of Sciences of the Czech Republic
616 62 Brno, Czech Republic
e-mail: kozak@ipm.cz

The base idea of this contribution is exact measurement of deformation and fracture material characteristics in laboratory, evaluation of this parameters and their application in models of finite element analysis modelling the fracture behaviour of components with defects. New approaches of the experimental and theoretical fracture mechanics are giving suitable tools. In difference of classical fracture mechanics describing more likely microscopically and global behaviour of materials up to date approaches are trying to predict behaviour of materials on the base of knowledge of micromechanics and micro mechanisms of damage.  Their typical range is problems connected with the so-called local approach. 

Toughness scaling model
The local approach for cleavage fracture is based on the weakest link concept that postulates that failure of the body of a material containing a large number of statistically independent volumes is triggered by the failure of one of the reference volume V0. The Beremin cleavage model is phrased within a two-parameter Weibull description of the failure probability by

    

The volume integral over the fracture process zone is denoted sw and is termed the Weibull stress. This stress is conventionally defined by

				       		

where m is so-called Weibull slope, V0 is a reference (or characteristic) volume, the integral is computed over the plastic zone, and s1 is the maximum principal stress. The parameters su and m of the Weibull stress sw at fracture are material parameters, i.e. independent of the stress state of materials, but may depend on the temperature. Local approach parameters for Beremin model were calibrated against laboratory data using two and three dimensional large strain elastic-plastic FEM analysis (2D for SSY condition, 3D for three point bending tests). Some results can be seen in the following figure.

GTN model
Using micromechanical data and FEM software WARP3D the dependence of J-integral on the ductile crack length has been received. As can be seen the computed curve and the experimental curve are nearly the same, but the good correlation has been found due to varying of q2 parameter. The values q1=1.5 and q2=1 have been accepted for a long time as constants non-dependent on the material behaviour.  In our calibration the q1 parameter is fixed and the dependence of q2 (representing the local triaxiality) on the slope of the R-curve is tested.  In connection of the slope the attention has been paid on the ductile crack initiation. From physical-mechanical point of view the value of Ji is appeared to be independent on the geometry of the body. This value can be regarded as a material characteristic and it has been determined numerically as the value when the first element vanished. This value can be denominate as a pseudo-physical and its dependence on the q2 parameter and on the triaxiality factor h has been determined. As can be seen the decreasing of the local stress triaxiality leads to the increasing of the initial value of Ji FEM and for our experimental steel this initiation value was markedly higher then for recommended value of q2. The value of Ji is dependent on the geometry of the body (from the engineering approach) and this finding in necessary to take into account in the case of transferability of the experimentally determined data to the real components.

Cohesive model for ductile materials
One of the key problems in the application of the cohesive model is the choice of the material law within the cohesive zone. For the determination of the cohesive stress, To in the case of normal fracture a hybrid technique has been developed. Using conventional elastic-plastic analysis, the distribution of the axial stress across the notch section of the specimen geometry is determined for the instant of the crack initiation in the centre of specimen. At that event, the axial stress exhibits a maximum in the centre of specimen, which is supposed to be equal to To. The cohesive energy, Go can be determined in a fracture mechanics test by assuming that Go equals the J-integral at initiation of stable crack extension, Ji. The procedure can be taken from the standard test methods. The stretch zone width at initiation is determined on the least three specimens exhibiting ductile tearing beyond the 0.2 mm offset line. The intersection point of the average SZWi and J-Da curve defines Ji. 
The present applications of cohesive models are still far away from practical engineering employment in structural integrity assessments. There is a strong need to standardize the simulation techniques and determination of the model parameters. New experimental test methods are required for especially for determining micro structural properties and for the calibration of numerical analysis.