PSI - Issue 32

N.V. Boychenko et al. / Procedia Structural Integrity 32 (2021) 326–333 Boychenko N.V./ Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction Assessment of influence of the crack tip shape and size on the fracture characteristics is an actual problem in fracture mechanics. In general, fracture in a brittle or ductile material occurs due to the micro crack initiation and growth caused by the high stress concentration. Basically, linear fracture mechanics addresses ideally-sharp crack problems or mathematical notch, when the crack tip radius equals to zero. In fact, the crack tip has a finite radius due to the blunting. The blunting occurs due to plastic deformations observed in the vicinity of the crack tip during the loading. Therefore, the originally blunted crack with a finite root radius is used in numerical simulation of McMeeking (1977) . O’Dowd and Shih (1991) found that stress-strain fields do not depend on the initial root radius when the crack tip blunted beyond about three times the initial root radius. Thus, the solution is independent of initial crack curvature radius and may be interpreted as those pertaining to initially sharp crack, except for the local area in the immediate vicinity of the blunting crack tip. The influence of the shape and size of crack tip for the fracture mechanics problems have enjoyed attention of researchers. It should be noted that most studies on effect of the crack root radius were carried out either in model problems of bodies of infinite size, or in experimental specimens containing a crack or notch with a root radius of the order of 100 μm or more. The strain gradient plasticity (SGP) theory has received significant attention, due to the possibility to relate fracture at the macro level with processes occurring at the micro- and mesoscale. A length scale parameter incorporated in SGP models allows predicting the scale effects observed in experiments at small scale levels. Thereby, it is necessary to simulate the crack with sizes corresponding to the dimensions of real cracks appearing in test specimens and structural components. Attention should be focused on understanding what crack root radius should be used in numerical analysis of fracture mechanics problems taking into account the processes occurring at the microlevel. In the present work the crack cuvature radius effect on the stress fields for the strain gradient plasticity in comparison with the classical Hutchinson-Rice-Rosengren (HRR) solution is investigated.

2. Stress-strain field near the crack tip 2.1. Hutchinson-Rice-Rosengren fields

Quantitative description of the stress-strain fields around the crack tip is the basis for failure prediction of structural components with defects. The fundamentals of the analytical study of the stress-strain state in the plasticity zone surrounding the crack tip for a strain-hardening material are established in the classical Hutchinson Rice-Rosengren model (Hutchinson J.W. (1968), Hutchinson J.W. (1968), Rice at al. (1968)). The HRR solution describes the stress, strain and displacement fields near the crack tip in the following form:

1/( 1) n

 

( , ) n

p K r





ij  

,

(1)

ij

/( 1) ( , ) n n n n    

,

(2)

p K r

  

ij

ij

1/( 1) ( , ) n n u n  

(3)

u K r  

i

p

i

where ( , n) ij   , ( , n) i u  are the components of tensors of dimensionless stresses, strains and displacements, respectively; K p is plastic stress intensity factor; a and n=1/N are material hardening parameters; N is strain hardening exponent, r and θ are polar coordinates centered on the crack tip. The established singularity type is among the advantages of the HRR model. However, HRR-solution does not take into account the blunting effect, since, according to the asymptotic structure, when 0  r the stresses aim to infinity. The constitutional equations of classical plasticity do not contain the length scale material parameter and, therefore, cannot predict the size effects observed at small scale levels. ( , n) ij   ,