Dimensional Analysis and Fractal Modeling of Fatigue Crack Growth

In the present paper, generalized Paris and Wohler equations are derived according to dimensional analysis and incomplete similarity concepts. They provide a rational interpretation to a majority of empirical power-law criteria used in fatigue. In particular, they are able to model the effects of the grain size, of the initial crack length, as well as of the size-scale of the tested specimen on the crack growth rate and on the fatigue life. Regarding the important issue of crack-size dependencies of the Paris’ coefficient C and of the fatigue threshold, an independent approach, based on the application of fractal geometry concepts, is proposed to model such an anomalous behavior. As a straightforward consequence of the fractality of the crack surfaces, the fractal approach provides scaling laws fully consistent with those determined from dimensional analysis arguments. The proposed scaling laws are applied to relevant experimental data related to the crack-size and to the structural-size dependencies of the fatigue parameters in metals and in quasi-brittle materials. Finally, paying attention to the limit points defining the range of validity of the classical Wohler and Paris power-law relationships, correlations between the so-called cyclic or fatigue properties are proposed, giving a rational explanation to the experimental trends observed in the material property charts.

[1]  Paolo Lazzarin,et al.  Fatigue Strength Assessments of Welded Joints: from the Integration of Paris’ Law to a Synthesis Based on the Notch Stress Intensity Factors of the Uncracked Geometries , 2008, CP 2013.

[2]  K. Tanaka,et al.  4.04 – Fatigue Crack Propagation , 2003 .

[3]  Pietro Cornetti,et al.  New unified laws in fatigue: From the Wohler's to the Paris' regime , 2007 .

[4]  Alberto Carpinteri,et al.  Strength and toughness in disordered materials: complete and incomplete similarity , 1994 .

[5]  F. Erdogan,et al.  The Effect of Mean Stress on Fatigue Crack Propagation in Plates Under Extension and Bending , 1967 .

[6]  Zdenek P. Bazant,et al.  Fatigue Fracture of High-Strength Concrete and Size Effect , 1993 .

[7]  G. Totten.  FATIGUE CRACK PROPAGATION , 2008 .

[8]  K. Chan,et al.  Scaling laws for fatigue crack , 1993, Metallurgical and Materials Transactions A.

[9]  V. M. Radhakrishnan,et al.  Parameter representation of fatigue crack growth , 1979 .

[10]  A. J. Mcevily,et al.  Prediction of the behavior of small fatigue cracks , 2007 .

[11]  H. Kitagawa,et al.  Fracture Mechanical Approach to Very Small Fatigue Crack Growth and to the Threshold Condition , 1979 .

[12]  Alberto Carpinteri,et al.  Plastic flow collapse vs. separation collapse (fracture) in elastic-plastic strain-hardening structures , 1983 .

[13]  Andrea Carpinteri,et al.  A fractal analysis of size effect on fatigue crack growth , 2004 .

[14]  L. Molent,et al.  Crack growth of physically small cracks , 2007 .

[15]  M. E. Haddad,et al.  Prediction of non propagating cracks , 1979 .

[16]  Alberto Carpinteri,et al.  A unified fractal approach for the interpretation of the anomalous scaling laws in fatigue and comparison with existing models , 2009 .

[17]  P. Paris.  A rational analytic theory of fatigue , 1961 .

[18]  Alberto Carpinteri,et al.  Size effect in fracture toughness testing: a dimensional analysis approach , 1980 .

[19]  O. Basquin.  The exponential law of endurance tests , 1910 .

[20]  Zhang Jun,et al.  Fatigue Performance in Flexure of Fiber Reinforced Concrete , 1998 .

[21]  Alberto Carpinteri,et al.  Fractal nature of material microstructure and size effects on apparent mechanical properties , 1994 .

[22]  A. Carpinteri.  Scaling laws and renormalization groups for strength and toughness of disordered materials , 1994 .

[23]  R. Ritchie,et al.  Propagation of short fatigue cracks , 1984 .

[24]  Alberto Carpinteri,et al.  Fractal and multifractal approaches for the analysis of crack-size dependent scaling laws in fatigue , 2009 .

[25]  Michele Ciavarella,et al.  One, no one, and one hundred thousand crack propagation laws: A generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth , 2008 .

[26]  Alberto Carpinteri,et al.  A dimensional analysis interpretation to grain size and loading frequency dependencies of the Paris and Wöhler curves , 2011 .

[27]  Michele Ciavarella,et al.  On the possible generalizations of the Kitagawa–Takahashi diagram and of the El Haddad equation to finite life , 2006 .

[28]  R. Forman,et al.  Numerical Analysis of Crack Propagation in Cyclic-Loaded Structures , 1967 .

[29]  Alberto Carpinteri,et al.  Notch sensitivity in fracture testing of aggregative materials , 1982 .

[30]  K. Miller.  THE SHORT CRACK PROBLEM , 1982 .

[31]  Andrea Spagnoli,et al.  Self-similarity and fractals in the Paris range of fatigue crack growth , 2005 .

[32]  M. Ciavarella,et al.  A generalized Paris' law for fatigue crack growth , 2006 .

[33]  Michael F. Ashby,et al.  Overview no. 112: The cyclic properties of engineering materials , 1994 .

[34]  Alberto Carpinteri,et al.  Self-similarity and crack growth instability in the correlation between the Paris’ constants , 2007 .

[35]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[36]  G. I. Barenblatt,et al.  INCOMPLETE SELF‐SIMILARITY OF FATIGUE IN THE LINEAR RANGE OF CRACK GROWTH , 1980 .

[37]  Zdenek P. Bazant,et al.  Size Effect in Fatigue Fracture of Concrete , 1991 .

[38]  Alberto Carpinteri,et al.  Static and energetic fracture parameters for rocks and concretes , 1981 .

[39]  Paul C. Paris,et al.  Service load fatigue damage — a historical perspective , 1999 .

[40]  D. S. Dugdale,et al.  The propagation of fatigue cracks in sheet specimens , 1958 .

[41]  L. Molent,et al.  Recent developments in fatigue crack growth assessment , 2006 .

[42]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[43]  N. Riley,et al.  Scaling, self-similarity, and intermediate asymptotics , 1998 .

[44]  Andrea Spagnoli,et al.  Fractality in the threshold condition of fatigue crack growth: an interpretation of the Kitagawa diagram , 2004 .

[45]  P. C. Paris,et al.  A Critical Analysis of Crack Propagation Laws , 1963 .

[46]  C. Kesler,et al.  Effect of Range of Stress on Fatigue Strength of Plain Concrete Beams , 1958 .