# Mathematical description of a ball hitting a glass plate and causing damage

In summary, when a ball hits a glass plate, it can cause damage to the plate and this damage can be represented by using the stress and strain tensors. The yield stress is the point at which the material begins to permanently deform, while the ultimate stress is the point at which the material breaks entirely. For brittle materials like glass, there is a small difference between these two stresses. The strength of the glass is also dependent on the initial flaw size, with a higher flaw size resulting in a lower breaking strength.
Summary: damage on glasses plate by hitting ball

If a ball hit a rectangular plate of glass and it cause a damage "but it didn't broken" how can i represent this damage using mathematics?
I'll tell you my idea about it: let the ball move horizontally and hit the plate perpendicularly with momentum mv then the damage will be described as stress or strain tensor. This my idea if it's wrong tell me the truth.

You are correct that the stress and strain tensors are used to represent the material properties of materials. What you are probably interested in is the yield stress. At this point the material will begin to permanently deform. The ultimate stress would be the stress at which it breaks entirely. For materials like steel there is a fairly large difference between the yield stress and the ultimate stress, but for brittle materials like glass they are fairly close.

Dale said:
You are correct that the stress and strain tensors are used to represent the material properties of materials. What you are probably interested in is the yield stress. At this point the material will begin to permanently deform. The ultimate stress would be the stress at which it breaks entirely. For materials like steel there is a fairly large difference between the yield stress and the ultimate stress, but for brittle materials like glass they are fairly close.
thank you for your response but can you give me a site to see an example?
another question: is yield stress the material not broken, ultimate stress the material is broken? like the body on rough plane and a force affect on it, if its not large enough to move the body the friction force not limiting, if its make it about to move the friction force is limiting? true?

Now that’s a question that would take a textbookCourse! In brief one can calculate the force on the glass plate knowing the Velocity of the ball and convert it to kinetic energy. The force, based on the stiffness, can be used to determine the contact stress and if high enough exceeding the strength of the glass it will cause a crack but it will not propagate through and break as it is very local.However at the back of the plate where the bending stress is in tension if that stress exceeds the ultimate then it will crack and break this time because it is not local anymore. The strength of the glass is dependent on the initial flaw size in the glass. It can vary anywhere between 1000 PSI and 40,000 PSI depending on the depth of the initial flaw. Glass, without the flaw, has a ridiculously high breaking strength.

PhanthomJay said:
Now that’s a question that would take a textbookCourse! In brief one can calculate the force on the glass plate knowing the Velocity of the ball and convert it to kinetic energy. The force, based on the stiffness, can be used to determine the contact stress and if high enough exceeding the strength of the glass it will cause a crack but it will not propagate through and break as it is very local.However at the back of the plate where the bending stress is in tension if that stress exceeds the ultimate then it will crack and break this time because it is not local anymore. The strength of the glass is dependent on the initial flaw size in the glass. It can vary anywhere between 1000 PSI and 40,000 PSI depending on the depth of the initial flaw. Glass, without the flaw, has a ridiculously high breaking strength.

PhanthomJay

## 1. What is the mathematical equation for describing a ball hitting a glass plate?

The mathematical equation for describing a ball hitting a glass plate is known as the impact force equation, which is F = m * v/t, where F is the force, m is the mass of the ball, v is the velocity, and t is the time of impact.

## 2. How does the angle of impact affect the damage caused by a ball hitting a glass plate?

The angle of impact plays a significant role in determining the damage caused by a ball hitting a glass plate. A ball hitting the glass plate at a perpendicular angle will cause more damage compared to a ball hitting the plate at an oblique angle. This is because a perpendicular impact transfers more energy to the glass plate, resulting in a higher force and more damage.

## 3. Is there a specific velocity at which a ball will shatter a glass plate upon impact?

Yes, there is a specific velocity at which a ball will shatter a glass plate upon impact. This velocity is known as the critical velocity and is dependent on factors such as the size and material of the ball and the thickness and type of glass in the plate. If the impact velocity exceeds the critical velocity, the glass plate will shatter.

## 4. Can the damage caused by a ball hitting a glass plate be predicted using mathematical models?

Yes, mathematical models can be used to predict the damage caused by a ball hitting a glass plate. These models take into account various factors such as the material properties of the ball and glass, the angle and velocity of impact, and the thickness of the glass plate. By inputting these variables into the model, scientists can accurately predict the extent of damage caused by the impact.

## 5. How can mathematical descriptions of ball-glass impact be used in real-world applications?

The mathematical description of a ball hitting a glass plate can be used in various real-world applications, such as designing safety glass for buildings and vehicles. By understanding the impact forces and damage caused by a ball hitting a glass plate, engineers can develop stronger and more durable glass materials to prevent shattering and potential injuries. This knowledge can also be applied in sports equipment design, where impact forces and angles are crucial in determining the safety and effectiveness of the equipment.

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