Permanent Deformation (PD): Phenomenological Model, Part I

Part of my graduate research study aimed to incorporate the shear stress component into various predictive models for permanent (plastic) deformation under cyclic loading in granular materials. In particular, granular materials referred, herein, are aggregate base and subgrade materials constructed under pavement or roadway subjected to repetitive vehicular loading as opposed to saturated sand and silt subjected to cyclic loading during seismic shaking events (i.e. undrained response). In that context, total stress was used instead of effective stress to simplify the problem. 

One should expect accumulation of permanent strain (or deformation) plotted as a function of load cycle (or time) as above. Note that the curve resembles a highly nonlinear relationship between plastic strain and load cycle; a phenomenological model that describes empirical relationship of each parameter. In one of the early studies, Monismith et. al. (1975) describes the power relationship to best fit the data by least sum of squared errors. Taking log at both sides, the power relationship becomes a linear relationship, shown in the figure below, where \(\epsilon_{p}\) = permanent strain; \(N\) = number of load cycle; \(A\) and \(b\) = experimentally determined coefficients.

$$\epsilon_{p} = AN^b$$

$$log(\epsilon_{p}) = log(A) + b\cdot log(N)$$

It now becomes obvious that coefficients \(A\) and \(b\), respectively, represent the "intercept" and "slope" of the data. The exponent \(b\) is a material constant; the exponent \(A\) must be a function of other factors such as stress level, previous stress history, placement conditions, etc. Furthermore, the exponent \(b\) is typically around 0.1, and \(A\) is harder to define due to other various factors. However, other studies have shown that \(A\) is strongly dependent on repeated stress state and material strength (Khedr, 1985; Garg et. al., 1997).

Using the laboratory test data obtained in my study, we will write a Python code to curve fit the data next.   

See Next: Curve Fitting with Python

Reference:

Gard, N, Tutumluer E, Thompson, MR, 1998. Structural modeling concepts for the design of airport pavements for heavy aircraft. Proceedings of the 5th International Conference on the Bearing Capacity of Roads and Airfields, Trondheim, Norway, 1998.

Khedr, S, 1985. Deformation characteristics of granular base course in flexible pavement. Transportation Research Record: Journal of the Transportation Research Board, 1043: 131-138.

Monosmith, CL, Ogawa, N,  Freeme, CR, 1975. Permanent deformation characteristic of subgrade soils due to repeated loading. Transportation Research Record: Journal of the Transportation Research Board, 537: 1-17.

Book Review: Zero to One

Summary

Peter Thiel, the co-founder of PayPal and Palantir, describes the ways to run successful startups. Thiel highlights how big progresses are leaped in the vertical direction, not horizontal. That is, entrepreneurs should focus on their unique identifier and make exponential growth out of it, but not incrementally improve existing products. Secrets are hard problems to solve. Conventions and mysteries are either too easy or impossible. The foundations are really important. Decisions made early on can be hard to change later. Early mistakes can prove fatal to startups. Find the right team. When you have the product, find a way to sell or distribute it. The core principles are summarized in the seven questions.

Two takeaways from the book:

• Monopolies are good for business: Competition destroys profits. Yep! That means you regulate the market that you own. Government, economists, competitors, and consumers do not like that. So, you will have to be sneaky and careful.     
• Ask yourself the seven questions. You must address every one of them. If you nail all seven, congrats! If you get five or six, it may work. These questions are:
1. Engineering: Can you create breakthrough vs. incremental improvements?
2. Timing: Is now the right time?
3. Monopoly: Start small and monopolize?
4. People: Do you have the right team?
5. Distribution: Do you have a way to not just create but deliver your product?
6. Durability: Will your market position be defensible 10 and 20 years later?
7. Secret: What's your secret recipe?

Who should read the book:

Anyone wants to follow the money.

* * *

I like how Theil starts from the core principles, then works toward the mechanics before deriding the cleantech bubble. The seven questions are gold. But a couple of those may not be applicable to business products vs. the consumer products that he has built. He is optimistic about the future for the sake of continuous innovation, but the rest of his advice about life was debatable. He is a contrarian in higher education, i.e. the Thiel Fellowship gives $100,000 to young entrepreneurs (age 22 or younger) specifically requires recipients to drop out of school. I don't know about that. This reminds me of the perception that education is worthless. Apparently, he is familiar with the Gaussian distribution (i.e. normal distribution, bell curve). However, he misses the point of where these "successful" or highly applaudable figures like Zuckerberg, Jobs, Thiel himself, etc. fall on the curve? Maybe towards the tail. Thiel appears to enjoy being an elite of Silicon Valley in the very last part of the book. His optimistic view of the future reminds me of Harari's book about superhumans. Now, I wonder if Harari thinks the way Thiel does. Although I disagreed with many of Thiel's points, I enjoyed the book though.