Bažant-Le-Kirane Paradox of fatigue failure in engineering materials

Mahendra Gattu

doi:10.1515/ijmr-2024-0038

Article

Bažant-Le-Kirane Paradox of fatigue failure in engineering materials

Mahendra Gattu

Published/Copyright: November 27, 2024

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From the journal International Journal of Materials Research Volume 115 Issue 11-12

Abstract

The problem of fracture in quasi-brittle materials is strongly linked with a zone of micro-cracking known as the fracture process zone (FPZ). For monotonic loading, the material length scale parameter D₀ is used in strength scaling law to describe the transition from strength criteria to linear elastic fracture mechanics criteria. The Paris Law approach to quasi-brittle materials for cyclic loading introduces another length scale parameter, D_0c. Experimental testing of concrete, rock, and sandstone showed two contradictory results. A few experiments showed that the parameter D_0c is higher than D₀, and the cyclic FPZ is larger than monotonic FPZ. Another set of experiments showed that D_0c is smaller than D₀, and the cyclic FPZ is smaller than monotonic FPZ. This interesting contradiction is named the Bažant-Le-Kirane Paradox (B-L-K Paradox) after the scientists involved in the experimentation. The B-L-K Paradox is the holy grail of fatigue fracture mechanics, and solving this problem will allow for the rational design of concrete structures.

Keywords: Cyclic fracture process zone; Monotonic fracture process zone; Fatigue; Fracture; Sandstone; Concrete

Corresponding author: Mahendra Gattu, Department of Civil Engineering, NIT-Rourkela, 769008, Rourkela (Odisha), India, E-mail: g2mahendra@gmail.com

Acknowledgements

The author acknowledges the financial support from NIT-Rourkela for attending the ICPCM-2023.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The author have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interests: The author state no conflict of interest.
Research funding: None declared.
Data availability: Not applicable.

Appendix: Dimensional analysis and incomplete similarity of Paris Law

Dimensional analysis has been adopted by scientists ¹⁵ for obtaining simple relationships between physical quantities measured in experiments. For instance, we have q₀ = F(q₁, q₂ … , q_n; s₁, s₂, … , s_m; r₁, r₂, … , r_m)

q₀ is the physical parameter of interest.

q_i are quantities with independent physical dimensions i.e. none of these quantities can be expressed as products of powers of the dimensions of remaining quantities.

s_j are quantities which can be expressed as products of powers of the dimensions of parameters q_i.

r_m are dimensionless quantities.

The crack growth equation d a d N is expressed as follows:

(4) d a d N = F ( σ y , K Ic , ω ; Δ K , Δ K th , σ Fl , E , h ; R )

Using the Buckingham π-theorem, we define π_J = ΔK/K_Ic, π₂ = ΔK_th/K_Ic, π₃ = Δσ_Fl/σ_y, π₄ = E/σ_y, π 5 = σ y 2 h / K Ic 2 , π 6 = σ Fl 2 a / Δ K th 2 , π₇ = 1−R.

The parameter π₅ is equal to h/r_p (also denoted by z²) and governs the transition from small-scale yielding to large-scale yielding.

For complete similarity,

(5) lim π 1 → 0 o r π 1 → ∞ ϕ ( π 1 , π 2 , π 3 , π 4 , π 5 , π 6 , π 7 ) = ϕ 1 ( π 2 , π 3 , π 4 , π 5 π 6 , π 7 )

In some cases: limϕ → 0 limϕ → ∞. This means the quantity π₁ is essential no matter how small or large it becomes and the function ϕ has a power-type asymptotic representation. We get incomplete similarity:

(6) lim π 1 → 0 o r π 1 → ∞ ϕ ( π 1 , π 2 , π 3 , π 4 , π 5 , π 6 , π 7 ) = ( π 1 ) β 1 ϕ 1 ( π z , π 3 , π 4 , π 5 π 6 , π 7 )

We have

(7) d a d N = ( K Ic σ y ) 2 ( Δ k K Ic ) β 1 ϕ 1 ( π z , π 3 , π 4 , π 5 , π 6 , π 7 )

This results in

(8) d a d N = K IC 2 − m σ y 2 ( π b r p ) β 2 ( π a a 0 ) β 3 ( 1 − R ) β 4 ϕ 2 ( Π 2 , π 3 , π 4 ) ( Δ K ) β 1

Here m = β₁, C = K IC 2 − m σ y 2 ( π b r p ) β 2 ( π a a 0 ) β 3 ( 1 − R ) β 4 ϕ 2 ( Π 2 , π 3 , π 4 ) . The constant C is dependent on K_IC, σ_y, R, h, a.

For incomplete similarity in π 5 = h r P = K Ic 2 σ y 2 h : If π₅ is neither too small or too large, i.e., the size of process zone is comparable to structure size, a transition from small-scale yielding to large-scale yielding occurs. C is structure-size dependent for incomplete self-similarity in π₅.

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Received: 2024-01-30

Accepted: 2024-04-09

Published Online: 2024-11-27

Published in Print: 2024-11-26

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Articles in the same Issue

https://doi.org/10.1515/ijmr-2024-0038

Keywords for this article

Cyclic fracture process zone; Monotonic fracture process zone; Fatigue; Fracture; Sandstone; Concrete