A New General Form of Weight-Function for One-Dimensional Edge Cracks: Application to Mode I Stress Intensity Factor Evaluation of a Radial Edge Crack in a Solid Circular Disk

Imagine a world made of giant, invisible Lego blocks. In this world, engineers build bridges, gears, and spinning disks that have to work perfectly for years. But sometimes, tiny cracks start to form, like a hairline fracture in a favorite ceramic mug. These cracks are dangerous because they can grow silently until the whole thing snaps. To stop this, scientists use a special tool called "fracture mechanics." Think of it as a crystal ball that looks at a crack and asks, "How bad is the stress right at the tip of this crack?" The answer to that question is called the Stress Intensity Factor. If this number gets too high, the crack will grow, and the machine will break.

For a long time, figuring out this number was like trying to solve a massive, complicated math puzzle every single time you wanted to check a crack. You had to build a new digital model of the object, simulate the forces, and crunch the numbers. It was slow, expensive, and took a lot of computer power. But there is a smarter way called the Weight Function. Imagine this as a "universal translator" or a master key. Once you have the right key for a specific shape (like a round disk), you can use it to instantly calculate the stress for any kind of load pushing on it, without having to rebuild the whole puzzle every time. The big question is: Can we make a better, more accurate key for these round disks with cracks?

This is exactly what Mohammad Mokarrami and Javad Alizadeh Kaklar set out to do in their paper. They are like master locksmiths trying to forge a new, super-precise key for a specific type of lock: a solid circular disk with a crack running from the edge toward the center.

In their study, the authors didn't just guess what the key should look like; they built a brand new mathematical formula, which they call a "general form of the weight function." Think of this formula as a new recipe for a magic sauce. To test if their recipe works, they needed to know exactly how much "stress" (or pressure) was hitting the crack in a perfect, computer-simulated world. They used a powerful software called Abaqus to run 18 different simulations, changing the length of the crack from very short (5% of the disk's width) to very long (90% of the width). For each of these 18 crack sizes, they applied three different types of "reference loads"—imagine pressing on the crack with a flat hand, a slanted hand, and a curved hand—to see how the crack reacted.

Once they had these reference numbers, they plugged them into their new recipe to find the secret ingredients, which they call "coefficients" (named $M_1$, $M_2$, and $M_3$). These numbers tell the formula exactly how to behave for any crack size. To make the tool even easier to use, they created a smooth, continuous curve (a fitting function) that lets engineers calculate these coefficients for any crack length between 0.05 and 0.9, not just the 18 specific ones they tested.

The results were impressive. To prove their new key worked, they tried it on five tricky, wiggly stress patterns—loads that change in complex, non-straight ways, like a sine wave or a sharp curve. When they compared their new formula's answers to the heavy-duty computer simulations, the results were almost identical. The difference was so tiny that the biggest mistake was only 0.8%, and the average mistake was just 0.2%. That's like measuring a football field and being off by less than the width of a fingernail.

They also compared their new formula to older, famous keys used by other scientists. They found that their new method was just as accurate as the most complex old methods, but it was much simpler. While some older formulas needed up to ten different secret numbers to work, their new one only needed three. And when they tested it against a known "exact" solution for a specific type of load, their new formula was incredibly precise, with an average error of less than 0.1%.

In short, the authors didn't just find a better way to measure stress; they created a streamlined, highly accurate, and easy-to-use tool. They showed that you don't need a massive, complicated computer simulation every time you want to check a crack in a spinning disk. Instead, you can use their new formula to get a near-perfect answer instantly. This means engineers can design safer, more reliable machines without spending days waiting for computer calculations, all while keeping the safety margins incredibly tight.