Which Cracks Should . . . To Make a Proper . . . How Cracks Grow: A . . . Case of Very Short Cracks How to Decide Which Cracks Practical Case of . . . Should be Repaired First: Empirical Dependence . . . Beyond Paris Law Theoretical Explanation of Scale Invariance: Main . . . How Can We Use . . . Empirical Formulas Home Page Title Page Edgar Daniel Rodriguez Velasquez 1 , 2 , Olga Kosheleva 3 , and Vladik Kreinovich 4 ◭◭ ◮◮ 1 Universidad de Piura in Peru (UDEP), edgar.rodriguez@udep.pe ◭ ◮ Departments of 2 Civil Engineering, 3 Teacher Education, and 4 Computer Science Page 1 of 31 University of Texas at El Paso, El Paso, Texas 79968, USA, edrodriguezvelasquez@miners.utep.edu Go Back olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
Which Cracks Should . . . To Make a Proper . . . 1. Which Cracks Should Be Repaired First? How Cracks Grow: A . . . • Under stress, cracks appear in constructions. Case of Very Short Cracks Practical Case of . . . • They appear in buildings, they appear in brides, they Empirical Dependence . . . appear in pavements, they appear in engines, etc. Beyond Paris Law • Once a crack appears, it starts growing. Scale Invariance: Main . . . • Cracks are potentially dangerous. How Can We Use . . . Home Page • Cracks in an engine can lead to a catastrophe. Title Page • Cracks in a pavement makes a road more dangerous ◭◭ ◮◮ and prone to accidents, etc. ◭ ◮ • It is therefore desirable to repair the cracks. Page 2 of 31 • In the ideal world, each crack should be repaired as Go Back soon as it is noticed. Full Screen • This is indeed done in critical situations. Close Quit
Which Cracks Should . . . To Make a Proper . . . 2. Which Cracks to Repair First (cont-d) How Cracks Grow: A . . . • Example: after each flight, the Space Shuttle was thor- Case of Very Short Cracks oughly studied and all cracks were repaired. Practical Case of . . . Empirical Dependence . . . • In less critical situations, for example, in pavement en- Beyond Paris Law gineering, our resources are limited. Scale Invariance: Main . . . • In such situations, we need to decide which cracks to How Can We Use . . . repair first. Home Page • We must concentrate efforts on cracks that, if unre- Title Page paired, will become most dangerous. ◭◭ ◮◮ • For that, we need to be able to predict how each crack ◭ ◮ will grow, e.g., in the next year: Page 3 of 31 – once we are able to predict how the current cracks Go Back will grow, Full Screen – we will be able to concentrate our limited repair resources on most potentially harmful cracks. Close Quit
Which Cracks Should . . . To Make a Proper . . . 3. To Make a Proper Decision, It Is Desirable to How Cracks Grow: A . . . Have Theoretically Justified Formulas Case of Very Short Cracks • Crack growth is a very complex problem, it is very Practical Case of . . . difficult to analyze theoretically. Empirical Dependence . . . Beyond Paris Law • So far, first-principle-based computer models have not Scale Invariance: Main . . . been very successful in describing crack growth. How Can We Use . . . • Good news is that cracks are ubiquitous. Home Page • There is a lot of empirical data about the crack growth. Title Page • Researchers have come up with empirical approximate ◭◭ ◮◮ formulas. ◭ ◮ • In the following, we will describe the state-of-the-art Page 4 of 31 empirical formulas. Go Back • However, purely empirical formulas are not always re- Full Screen liable. Close Quit
Which Cracks Should . . . To Make a Proper . . . 4. Need for Theoretical Formulas (cont-d) How Cracks Grow: A . . . • There have been many cases when an empirical formula Case of Very Short Cracks turned out to be: Practical Case of . . . Empirical Dependence . . . – true only in limited cases, Beyond Paris Law – and false in many others. Scale Invariance: Main . . . • Even the great Newton naively believed that: How Can We Use . . . Home Page – since the price of a certain stock was growing ex- ponentially for some time, it will continue growing, Title Page – so he invested all his money in that stock and lost ◭◭ ◮◮ almost everything when the bubble collapsed. ◭ ◮ Page 5 of 31 Go Back Full Screen Close Quit
Which Cracks Should . . . To Make a Proper . . . 5. Need for Theoretical Formulas (cont-d) How Cracks Grow: A . . . • From this viewpoint: Case of Very Short Cracks Practical Case of . . . – taking into account that missing a potentially dan- Empirical Dependence . . . gerous crack can be catastrophic, Beyond Paris Law – it is desirable to have theoretically justified formu- Scale Invariance: Main . . . las for crack growth. How Can We Use . . . • This is what we do in this talk: we provide theoretical Home Page explanations for the existing empirical formulas. Title Page • With this goal in mind, let us recall the main empirical ◭◭ ◮◮ formulas for crack growth. ◭ ◮ Page 6 of 31 Go Back Full Screen Close Quit
Which Cracks Should . . . To Make a Proper . . . 6. How Cracks Grow: A General Description How Cracks Grow: A . . . • In most cases, stress comes in cycles: Case of Very Short Cracks Practical Case of . . . – the engine clearly goes through the cycles, Empirical Dependence . . . – the road segment gets stressed when a vehicle passes Beyond Paris Law through it, etc. Scale Invariance: Main . . . • So, the crack growth is usually expressed by describing: How Can We Use . . . Home Page – how the length a of the crack changes Title Page – during a stress cycle at which the stress is equal to some value σ . ◭◭ ◮◮ • The increase in length is usually denoted by ∆ a . ◭ ◮ Page 7 of 31 • So, to describe how a crack grows, we need to find out how ∆ a depends on a and σ : ∆ a = f ( a, σ ) . Go Back Full Screen Close Quit
Which Cracks Should . . . To Make a Proper . . . 7. Case of Very Short Cracks How Cracks Grow: A . . . • The first empirical formula – known as W¨ ohler law – Case of Very Short Cracks was proposed to describe how cracks appear. Practical Case of . . . Empirical Dependence . . . • In the beginning, the length a is 0 (or very small). Beyond Paris Law • So the dependence on a can be ignored, and we have Scale Invariance: Main . . . ∆ a = f ( σ ) , for some function f ( σ ). How Can We Use . . . Home Page • Empirical data shows that this dependence is a power law: ∆ a = C 0 · σ m 0 , for some constants C 0 and m 0 . Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 31 Go Back Full Screen Close Quit
Which Cracks Should . . . To Make a Proper . . . 8. Practical Case of Reasonable Size Cracks How Cracks Grow: A . . . • In critical situations, the goal is to prevent the cracks Case of Very Short Cracks from growing. Practical Case of . . . Empirical Dependence . . . • In such situations, very small cracks are extremely im- Beyond Paris Law portant. Scale Invariance: Main . . . • In most other practical viewpoint, small cracks are usu- How Can We Use . . . ally allowed to grow. Home Page • So the question is how cracks of reasonable size grow. Title Page • Several empirical formulas have been proposed. ◭◭ ◮◮ • In 1963, P. C. Paris and F. Erdogan compared all these ◭ ◮ formulas with empirical data. Page 9 of 31 • They came up with a new empirical formula that best Go Back fits the data: ∆ a = C · σ m · a m ′ . Full Screen • This Paris Law (aka Paris-Erdogan Law ) is still in use. Close Quit
Which Cracks Should . . . To Make a Proper . . . 9. Usual Case of Paris Law How Cracks Grow: A . . . • Usually, m ′ = m/ 2, so ∆ a = C · σ m · a m/ 2 = C · ( σ ·√ a ) m . Case of Very Short Cracks • Paris formula is empirical, but the dependence m ′ = Practical Case of . . . Empirical Dependence . . . m/ 2 has theoretical explanations. Beyond Paris Law • One of such explanations is that the stress acts ran- Scale Invariance: Main . . . domly at different parts of the crack. How Can We Use . . . Home Page • According to statistics, on average, the effect of n in- dependent factors is proportional to √ n . Title Page • A crack of length a consists of a/δ a independent parts. ◭◭ ◮◮ • So, the overall effect K of the stress σ is proportional ◭ ◮ to K = σ · √ n ∼ σ · √ a. Page 10 of 31 • This quantity K is known as stress intensity . Go Back • For the power law ∆ a = C · K m , this leads to ∆ a = Full Screen const · ( σ · √ a ) m = const · σ m · a m/ 2 , i.e., to m ′ = m/ 2. Close Quit
Which Cracks Should . . . To Make a Proper . . . 10. Empirical Dependence Between C and m How Cracks Grow: A . . . • In principle, we can have all possible combinations of Case of Very Short Cracks C and m . Practical Case of . . . Empirical Dependence . . . • Empirically, however, there is a relation between C and Beyond Paris Law m : C = c 0 · b m 0 . Scale Invariance: Main . . . How Can We Use . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 31 Go Back Full Screen Close Quit
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