The influence of the crack fractal geometry on Elastic-Plastic Frature Mechanics

1. The influence of the crack fractal geometry on Elastic-Plastic Frature Mechanics • Lucas Maximo Alves, DEMA-UEPG-PR, lumaximo@if.sc.usp.br; • Rosana Vilarim da Silva, EESC-USP Sao Carlos, rosavi@sc.usp.br • Bernhard Joachim Mokross, IFSC-USP Sao Carlos, mokross@if.sc.usp.br

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

3. Classical Fracture Mechanics (plane fracture surface) + Stable or quasistatic process (K, KIC, J-R-curve) Unstable or non-linear dynamics process (K(t), KID, JD-Rv-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 Application to the Fracture Mechanics

4. Fracture Phenomenon • Griffith-Irwin Balance Xdu = dU + JdL0 • Fixed grips conditions Xdu = 0 • Stable crack propagation J = dU/dL0 • Irwin-Orowan insight 2eff = 2e + p

5. The Fractal Model • Fracture surface as self-affine fractal • Sand-Box Method (balls recovering] used as instantaneous characterization can describe the crack propagation  = Lo/l0 Seed with random shape but with projected size l0

6. Ruggednes Considerations • Self-affinity • Perpendiculars directions have the same nature of the measure • Initial boxe of counting is square with size l0 • Ruggedness description

7. Energy to create two new surfaces • Classical Energy • Modern Fractal Energy

8. Resistance to Crack Propagation • Classical Crack Resistance • Modern Fractal Resistance

9. Griffith-Irwin-Orowan conditionto stable crack propagation • Classical Criterion • Modern Fractal Criterion

10. J-R Relations • Classical definition of J0 • Relationship between crack resistance and crack geometry

11. 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, lo, 2eff = 2e + p ) • Fitting Calculations and Plots (Theoretical J-R curves)

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

13. A1CT2 Sample • Fractal Fracture surface

14. A1CT2 Sample • J-R-curve fitting • H = 0,38(teo) • l0 = 0,12 • 2eff = 73,0

15. B2CT2 Sample • Fractal Fracture Surface

16. B2CT2 Sample • J-R curve J-R-curve fitting • H = 0,569(teo) • l0 = 0,075 • 2eff = 37,07

17. Chelidze proposition Mu and Lung Passoja and Mandelbrot G0 = G (L/L0) G0 = 2eff(^1-D) G0 = E loD Comparison with others models of the literature

18. 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

19. 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