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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,

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

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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

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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

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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

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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

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Energy to create two new surfaces Classical Energy Modern Fractal Energy

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Resistance to Crack Propagation Classical Crack Resistance Modern Fractal Resistance

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

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

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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)

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

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A1CT2 Sample Fractal Fracture surface

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

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B2CT2 Sample Fractal Fracture Surface

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

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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

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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

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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

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