Dimensional Analysis of Critical Phenomena: Self-Weight Failure, Turbulence, Resonance, Fracture
Source abstract
Abstract
Both in physics and engineering, Buckingham’s Π theorem is considered as a key tool for dimensional analysis, providing a method to identify the dimensionless parameters governing physical similitude and scale modeling, when the form of the solving equations is still unknown. In the present paper, the scaling of different critical phenomena in solid and fluid mechanics is emphasized by the application of the Π theorem. In particular, self-weight failure, turbulence, resonance, and fracture are considered, highlighting how these critical phenomena are governed by specific dimensionless numbers, which are functions of few fundamental mechanical quantities including the size-scale, which are characterized by different and algebraically independent physical dimensions.
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OpenAlex topics: High-Velocity Impact and Material Behavior · Landslides and related hazards · Rock Mechanics and Modeling
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References (37)
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1. Carpinteri, A., Size Effect in Fracture Toughness Testing: A Dimensional Analysis Approach, in Proceedings of the International Conference on Analytical and Experimental Fracture Mechanics, Roma, Italy, 1980, pp. 785–797.10.1007/978-94-009-8594-0_60 ↗
2. Carpinteri, A., A Fracture Mechanics Model for Reinforced Concrete Collapse, in Proceedings of the IABSE Colloquium on Advanced Mechanics of Reinforced Concrete, Delft, 1981, pp. 17–30.
3. Carpinteri, A., Notch Sensitivity in Fracture Testing of Aggregative Materials, Eng. Fract. Mech., 1982, vol. 16, pp. 467–481.10.1016/0013-7944(82)90127-8 ↗
4. Carpinteri, A., Stability of Fracturing Process in RC Beams, J. Struct. Eng. ASCE, 1984, vol. 110, pp. 544–558.10.1061/(ASCE)0733-9445(1984)110:3(544) ↗
5. Carpinteri, A., Interpretation of the Griffith Instability as a Bifurcation of the Global Equilibrium, in Application of Fracture Mechanics to Cementitious Composites. Proceedings of a NATO Advanced Research Workshop, Evanston, USA, 1984, Shah, S.P., Ed., Dordrecht: Martinus Nijhoff Publishers, 1985, pp. 287–316.10.1007/978-94-009-5121-1_10 ↗
6. Carpinteri, A., Cusp Catastrophe Interpretation of Fracture Instability, J. Mech. Phys. Solids, 1989, vol. 37, pp. 567–582.10.1016/0022-5096(89)90029-X ↗
7. Carpinteri, A., Structural Mechanics: A Unified Approach, London: Chapman & Hall, 1997.
8. Carpinteri, A., Cornetti, P., and Puzzi, S., Scaling Laws and Multiscale Approach in the Mechanics of Heterogeneous and Disordered Materials, ASME Appl. Mech. Rev., 2006, vol. 59, pp. 283–305.10.1115/1.2204076 ↗
9. Tipper, C., Brittle Fracture of Mild Steel Plates, in British Iron and Steel Research Association Proceedings, Cambridge, 1945, pp. 23–50.
10. Tipper, C., The Brittle Fracture Story, Cambridge: Cambridge University Press, 1962.
11. Barenblatt, G.I., Flow, Deformation and Fracture, Cambridge: Cambridge University Press, 2014.10.1017/CBO9781139030014 ↗
12. Barenblatt, G.I. and Botvina, L.R., A Note Concerning Power-Type Constitutive Equations of Deformation and Fracture of Solids, Int. J. Eng. Sci., 1982, vol. 20, pp. 187–191.10.1016/0020-7225(82)90015-5 ↗
13. Carpinteri, A., Scaling Laws and Renormalization Groups for Strength and Toughness of Disordered Materials, Int. J. Solids Struct., 1994, vol. 31, pp. 291–302.10.1016/0020-7683(94)90107-4 ↗
14. Carpinteri, A. and Pugno, N., Are Scaling Laws on Strength of Solids Related to Mechanics or to Geometry? Nature Mater., 2005, vol. 4, pp. 421–423.10.1038/nmat1408 ↗
15. Carpinteri, A. and Accornero, F., Rotation versus Curvature Fractal Scaling in Bending Failure, Phys. Mesomech., 2019, vol. 22, no. 1, pp. 46–51. doi 10.1134/S102995991901008910.1134/S1029959919010089 ↗
16. Goldstein, R.V. and Vainshelbaum, V.M., Material Scale Length as a Measure of Fracture Toughness in Fracture Mechanics of Plastic Materials, Int. J. Fracture, 1978, vol. 14, pp. 185–201.10.1007/BF00032544 ↗
17. Wilson, K.G., Renormalization Group and Critical Phenomena, Phys. Rev. B, 1971, vol. 4, pp. 3174–3183.10.1103/PhysRevB.4.3174 ↗
18. Panin, V.E., Overview on Mesomechanics of Plastic Deformation and Fracture of Solids, Theor. Appl. Fract. Mech., 1998, vol. 30(1), pp. 1–11.10.1016/S0167-8442(98)00038-X ↗
19. Panin, V.E., Synergetic Principles of Physical Mesomechanics, Theor. Appl. Fract. Mech., 2001, vol. 37(1–3), pp. 261–298.10.1016/S0167-8442(01)00085-4 ↗
20. Panin, V.E. and Egorushkin, V.E., Deformable Solid as a Nonlinear Hierarchically Organized System, Phys. Mesomech., 2011, vol. 14, no. 5–6, pp. 207–223.10.1016/j.physme.2011.12.002 ↗
21. Panin, V.E., Korotaev, A.D., Makarov, P.V., and Kuznetsov, V.M., Physical Mesomechanics of Materials, Russ. Phys. J., 1998, vol. 41(9), pp. 856–884.10.1007/BF02508720 ↗
22. Carpinteri, A. and Paggi, M., Singular Harmonic Problems at a Wedge Vertex: Mathematical Analogies Between Elasticity, Diffusion, Electromagnetism, and Fluid Dynamics, J. Mech. Mater. Struct., 2011, vol. 6, pp. 1–4.10.2140/jomms.2011.6.113 ↗
23. Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Equation, Phys. Rev., 1914, vol. 4(4), pp. 345–376.10.1103/PhysRev.4.345 ↗
24. Bertrand, J., Sur l’homogénéité dans les formules de physique, Comptes Rendus, 1878, vol. 86(15), pp. 916–920.
25. Vaschy, A., Sur les lois de similitude en physique, Annales Télégraphiques, 1892, vol. 19, pp. 25–28.
26. Federman, A., On Some General Methods of Integration of First-Order Partial Differential Equations, Proc. Saint-Petersburg Polytechnic Institute, 1911, vol. 16(1), pp. 97–155.
27. Riabouchinsky, D.P., Méthode des variables de dimension zéro et son application en aérodynamique, L’Aérophile, 1911, pp. 407–408.
28. Birkhoff, G., Hydrodynamics: A Study in Logic, Fact, and Similitude, Princeton University Press, 1956.
29. Borodich, F.M., Scaling Transformations in Solid Mechanics, in IUTAM Symposium on Scaling in Solid Mechanics, Springer, 2009, pp. 11–26.10.1007/978-1-4020-9033-2_2 ↗
30. Borodich, F.M., History of the Π Theorem, in Advances in Applied Mechanics, Elsevier, 2014, pp. 283–284.
31. Bridgman, P.W., Dimensional Analysis, New Haven: Yale University Press, 1931.
32. Curtis, W.D., Logan, J.D., and Parker, W.A., Dimensional Analysis and the Pi Theorem, Linear Algebra and Its Applications, 1982, vol. 47, pp. 117–126.10.1016/0024-3795(82)90229-4 ↗
33. Galilei, G., Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica e i movimenti locali, Leiden, The Netherlands: Ludovico Elzeviro, 1638.
34. Carpinteri, A., Mechanical Damage and Crack Growth in Concrete: Plastic Collapse to Brittle Fracture, Dordrecht: Martinus Nijhoff Publishers, 1986.10.1007/978-94-009-4350-6 ↗
35. Reynolds, O., An Experimental Investigation of the Circumstances which Determine whether the Motion of Water Shall be Direct or Sinuous, and of the Law of Resistance in Parallel Channels, Philos. Trans. Roy. Soc., 1883, vol. 174, pp. 935–982.10.1098/rstl.1883.0029 ↗
36. Kolmogorov, A.N., Dissipation of Energy in Locally Isotropic Turbulence, Proc. USSR Acad. Sci., 1941, vol. 32, pp. 16–18.
37. Timoshenko, S.P., Vibration Problems in Engineering, Van Nostrand, 1928.