Hi Abdolrasoul,

Your question is quite interesting. I will suggest an initial issue. Equations such as those you indicate are the result of combining empirical results with specific structural (e.g., plastic, elasto-plastic, elastic), thermodynamic, or other mechanisms.). Thus, epistemic uncertainty certainly plays a role in their development. specifically, the coefficients are estimated statistically through methods such as the maximum likelihood, the ordinary least squares, and can also be estimated using Bayesian regressions. Thus you have theoretical mechanisms that characterize the type of failure mechanisms conditioned on the type of material that is used in the design of your structure and includes empirical data regarding the performance of that material under a variety of stresses. I would suggest that it is not a matter of an equation being "reliable". Rather, it is a matter of deciding which equation or equations are appropriate to your specific project (which is not identified). Once the specific project is identified, the literature can guide you towards the most appropriate model to use. I suspect that the treatment of the uncertainty is normally probabilistic in the sense of frequencies (it is empirical in that sense), rather than Bayesian. An improvement along the lines of your question (reliability) seems to point to Bayesian methods, My view is that there is some conflation between the accuracy of the models themselves and the analysis of the uncertainty when parameterizing those models' coefficients (statistically speaking, those coefficients are parameters).

I hope I have understood your query.

All the best,

Paolo

Hi Paolo

Thank you for your interesting answer. You described well the current state of the art of the theory for analysis of failure of structures with defects. As you noted the current equations are the result of regression. In current regression methods you should select a generalized equation (1st source of epistemic uncertainty). You must have a set of reliable data (1st aleatory uncertainty). One use best fit the selected equation on the reliable data (2nd epistemic uncertainty). Conventional methods , cannot consider the expert will, e.g. the fragility curve should be higher than all sample data! (3rd source of epistemic uncertainty). These permanent uncertainties along with the probable error in numerical calculations led to different equations for the same problem! Clearly different solution for the same problem is not reliable. These uncertainties led the Griffith to present two solutions (one twice of the other) in 1920 and 1924 respectively, for the same problem!

Hi every one

Fracture mechanics is based on the following equation (1) at crack tip.

S=(K/sqrt(pi*r))*f+T (1)

in which k is stress intensity factor and r is radius from crack tip. Sources of uncertainty are as follows. SOR1: at crack tip r=0 then S=infinite! (which is impossible). SOR2: stress is defined as a ratio, so the denominator and numerator can be equally multiplied by a number! That is the K is not a material property. SOR3: The behavior is assumed as linear elastic, which is wrong at cracking conditions!

These sources of uncertainty made the fracture mechanics an unreliable field!

Consequently the conventional Failure Assessment Diagram (FAD) which is based on conventional fracture mechanics is also unreliable. The diversity in the proposed equations for (FAD) in the literature verify the above reasoning.

Dear Abdolrasoul Ranjbaran ,

Excellent statement!

That's why we are strictly avoid K-factors (SIF) and other non-energetic metrics for failure analysis.

Hi Martin Brandtner-Hafner

I appreciate your response.

Based on logical reasoning, it is found that when the crack depth ratio became 0.5 the cracked structure becomes unstable and fails!

Do you agree with this finding?

For ratio=0.5 the released energy density becomes equal to available energy density at the crack tip.

Perhaps a reason for verification of the above conclusion is the restriction of ratio to less than 0.5 for all available crack compliance’s in the literature.

The failure assessment diagram (FAD) is not meant as an exact theory. It is an engineering tool to connect in a practical way all failures of a defect containing structure ranging from crack tip controlled fractures (based on stress intensity) on the one end to structure controlled plastic collapse (based on limit analysis) on the other end. A material model is not an exact theory. It is only a model of the most important material properties for a considered problem. Besides neglecting less important aspects of material behavior you have the uncertainty of data.

The discussion here was on the stress intensity side. For the side of limit analysis I could point to two of our papers which contain sophisticated models, FEM analyses and hundreds of burst tests so that you can see the uncertainty at work:

Article Limit analysis of flaws in pressurized pipes and cylindrical...

Article Limit analysis of flaws in pressurized pipes and cylindrical...

A new perspective for the (FAD) is proposed in the attached paper. What is your idea regarding that?

https://thescipub.com/abstract/sgamrsp.2021.68.81

Good question Abdolrasoul Ranjbaran

Dear Abdolrasoul,

some comments about your SORs:

a) SOR1: obviously, that provides from a mathematical solution and a model that does not consider damage or plasticity at the crack tip. However, as long as such damage or plastic zone is sufficiently small, K describes the stress state and linear-elastic fracture mechanics provide good predictions;

b) SOR3: you should apply elastic-plastic fracture mechanics in non-linear situations.

As Prof. Staat states, FADs are an engineering tool, and do not intend to be absolutely precise for every situation. For example, considering that two specimens of the same geometry of material may fail under rather different loads (steels within the DBTZ), deterministic FADs cannot foreseen such differences and statistics should be included somewhere.

Concerning the best equation, the exact solution (Option 3 in some procedures) is the square root of the elastic J to total J ratio. "Lower" options are approximations based on the tensile properties. Such approximations are converging and to my understanding R6, BS7910 or FITNET provide the same equations.

Overall, in my experience with FADs, they are a very valuable engineering tool defined to derive generally conservative estimations of fracture loads.