The influence of the crack fractal geometry on Elastic-Plastic Frature Mechanics Lucas Maximo Alves, DEMA-UEPG-PR, l Rosana Vilarim da Silva, EESC-USP Sao Carlos, Bernhard Joachim Mokross, IFSC-USP Sao Carlos,

Introduction Fractal Theory Fractal Growth of Structures (Thermodynamics) Fractal Physics (Statistical Mechanics) Non-Linear Dynamics (Chaos Theory)

Application to the Fracture Mechanics Classical Fracture Mechanics (plane fracture surface) + Stable or quasistatic process (K, K IC, J-R- curve) Unstable or non- linear dynamics process (K(t), K ID, J D - R v -curve) Fractal Growth (ruggedness frature surface) Infuence of the crack fracture geometry on stable process Influence of the crack fracture geometry on the instability process

Fracture Phenomenon Griffith-Irwin Balance Xdu = dU + JdL 0 Fixed grips conditions Xdu = 0 Stable crack propagation J = dU/dL 0 Irwin-Orowan insight 2  eff = 2  e +  p

The Fractal Model Fracture surface as self-affine fractal Sand-Box Method (balls recovering] used as instantaneous characterization can describe the crack propagation Seed with random shape but with projected size l 0  = Lo/l 0

Ruggednes Considerations Self-affinity Perpendiculars directions have the same nature of the measure Initial boxe of counting is square with size l 0 Ruggedness description

Energy to create two new surfaces Classical Energy Modern Fractal Energy

Resistance to Crack Propagation Classical Crack Resistance Modern Fractal Resistance

Griffith-Irwin-Orowan condition to stable crack propagation Classical Criterion Modern Fractal Criterion

J-R Relations Classical definition of J 0 Relationship between crack resistance and crack geometry

Experimental Procedure J-R testing (Experimental J-R-curve) Scanning Eletronic Microscopy of fracture surface Image processing and analysis of Fractography Fractal characterizing of the fracture surface (H = 2 - D, l o, 2  eff = 2  e +  p ) Fitting Calculations and Plots (Theoretical J-R curves)

Results Fracture surface (J-R testing) Fractal dimension D and H = 2-D (from the Fractography) J-R - curves (fitting)

A1CT2 Sample Fractal Fracture surface

A1CT2 Sample J-R-curve fitting H = 0,38(teo) l 0 = 0,12 2  eff = 73,0

B2CT2 Sample Fractal Fracture Surface

B2CT2 Sample J-R curve J-R-curve fitting H = 0,569(teo) l 0 = 0,075 2  eff = 37,07

Comparison with others models of the literature Chelidze proposition Mu and Lung Passoja and Mandelbrot G 0 = G (L/L 0 ) G 0 = 2  eff (  ^1-D) G 0 = E l o D

Discussions This model is formally correct and it is non-linear Others authors have used the self-similar limit compromising the experimental results Alls the influences leave traces on the morphology of the fractured surface guaranting the success of the model

Conclusions The ruggedness of fracture surface explain the rising of the J-R curve The self-affine consideration fit the results better than the self-similar. The power law can be originating from the hardening We have a new fracture property more consistent We have a new method to obtain the J-R curve