1.1 - Formula for volumetric strain
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Introduction to Volumetric Strain
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Welcome, everyone! Today we are discussing local volumetric strain. Can anyone explain what happens to the volume of a body when it is deformed?
The volume changes, right? A part of the body may squeeze or stretch.
Exactly! That change in volume per unit volume gives us the concept of volumetric strain. It measures how much the volume of a local element changes during deformation.
How do we calculate that change?
Great question! We can calculate it using the scalar triple product, which is the determinant of a matrix formed by the vectors representing the sides of a small volume element.
So it’s like a 3D version of calculating area?
Yes, exactly! Just like we find area using two vectors, we find volume using three. And the formula allows us to express it mathematically.
To remember this, think of 'V for Volume' which relates back to 'D for Determinant'.
In summary, the volumetric strain gives us critical insight into how materials behave under stress, and it's unique for each point of interest.
Derivation of the Formulas
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Alright! Now, let's derive the formula for volumetric strain together. First, can someone remind us how we express the volume in reference configuration?
Isn't it V equals the scalar triple product, like V = ∆X · (∆Y × ∆Z)?
Perfect! Now, after deformation, we express the volume in the deformed state as v = ∆x · (∆y × ∆z).
And how do we relate this back to volumetric strain?
To find the volumetric strain, we take the difference between these two volumes and divide by the original volume V. The formula we derive simplifies down to showing that the volumetric strain is actually related to the trace of the displacement gradient matrix.
Wait, I remember something! The trace means we just need the diagonal elements?
Exactly! Good recall! This means regardless of how we choose our line elements, volumetric strain will remain the same at a given point.
In conclusion, the formula we derive for volumetric strain is critical for understanding material behavior when subjected to various loads.
Comparing Strain Types
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Let's wrap up by comparing volumetric strain with longitudinal and shear strains. Who can explain the difference?
Volumetric strain is about volume change, while longitudinal strain deals with length and shear strain changes the angles between lines.
Great! Volumetric strain is unique because it doesn't depend on how we choose our elements. Meanwhile, longitudinal and shear strains can vary based on that choice.
So, volumetric strain represents a kind of 'global' property in a localized area?
Yes! You can think of it as a more stable measure since it reflects a unique characteristic at any point of interest.
In short, understanding these differences helps engineers decide how to apply forces correctly in materials under various conditions. Remember: Volume is key!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section provides an overview of local volumetric strain, explaining how to define and calculate it using the scalar triple product and determinants. It emphasizes the uniqueness of volumetric strain at a point, independent of the choice of line elements.
Detailed
Detailed Summary
In this section, we delve into the concept of local volumetric strain that occurs when a body is deformed. As the body changes shape, the volume of local volume elements also changes. The local volumetric strain is defined as the change in volume per unit volume and varies in different parts of the body. To mathematically express this, we consider three line elements that form a parallelopiped in the reference configuration. The volume of the parallelopiped is given by the scalar triple product of the vectors representing its sides, and it can also be related to the determinant of a matrix formed by these vectors. Through derivation, we arrive at an expression for volumetric strain that highlights it as a unique property at a point, differentiating it from other strain types which can depend on the choice of line element triplets.
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Scalar Triple Product and Determinant Representation
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Chapter Content
The scalar triple product can also be realized as the determinant of a matrix whose columns are formed by the vectors involved in the scalar triple product. Thus, we can write...
Detailed Explanation
In this chunk, the text introduces the concept of the scalar triple product, which is an important mathematical construct for calculating the volume of a parallelopiped formed by three vectors. This scalar triple product can also be expressed as the determinant of a matrix that consists of these vectors as its columns. The scalar triple product provides a measure of the signed volume of the figure defined by the three vectors. This foundational formula allows us to connect the scalar triple product to the concept of volumetric strain.
Examples & Analogies
Imagine unscrewing a jar lid. The way the three edges of the jar lid move apart can be represented by three vectors. The 'space' these vectors define, like the volume in the jar before you open it, can be calculated using the scalar triple product. The determinant essentially measures how much 'space' you have between those edges.
Substituting and Simplifying the Formula
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Substituting the below formula in (6) leads to...
Detailed Explanation
In this part, the text describes how to substitute a specific formula into another one to simplify the expression for volumetric strain. The original formula is manipulated algebraically to arrive at a more convenient form, which is crucial for practical calculations in mechanics. The ongoing substitutions lead to the final formula that expresses volumetric strain in terms of easily measurable quantities.
Examples & Analogies
Think of making a smoothie. You start with the raw ingredients (like fruits and yogurt), which can be considered the original formula. As you blend, you mix and change those ingredients into a smooth drink (the final formula), making it easier to consume and enjoy, just as we simplify equations to make them easier to work with in mechanics.
Determinants and Higher-order Terms
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To obtain its determinant, let us expand it in terms of the first row...
Detailed Explanation
This section explains how to compute the determinant of the deformation gradient matrix by expanding it along the first row. The determinants are key in understanding the geometry of the deformation of materials. Here, the mention of higher-order terms is significant as it highlights that for small displacements, these terms contribute little to the calculations and can often be ignored.
Examples & Analogies
Imagine you are analyzing a small piece of rubber. If you stretch it just a little, the tiny changes (higher-order terms) in its shape are so minimal compared to the overall stretching. In math terms, just like ignoring the negligible adjustments makes our calculations cleaner and simpler, in physical terms, it helps us focus on the main changes happening to the rubber.
Final Formula for Local Volumetric Strain
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Finally, using equation (12), we get the following formula for local volumetric strain...
Detailed Explanation
Here, we arrive at the final expression that defines local volumetric strain. This formula is important because it links the strain to the properties of the material under slight deformations. The emphasis on its independence from the specific lines chosen illustrates that volumetric strain is a unique quantity characteristic of the material state at a point.
Examples & Analogies
Consider filling a balloon with water. No matter how you squeeze the balloon, the amount of water you can fit inside (volumetric strain) depends only on the volume of water, not how you manipulate the balloon. This represents the unique nature of volumetric strain at a point, independent of how you apply pressure.
Trace and Unique Nature of Volumetric Strain
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We can notice that the RHS of this equation is equal to the trace of the displacement gradient matrix...
Detailed Explanation
This chunk identifies that the right-hand side (RHS) of the derived equation corresponds to the trace of the displacement gradient matrix, reinforcing the mathematical underpinnings of volumetric strain. Understanding the trace in this context is valuable because it summarizes the effects of strain and ties together different aspects of the deformation in a concise manner.
Examples & Analogies
Think of the trace as a summary of all the notes you take during a class. Just as the trace boils down complex information into a simple, understandable summary, the trace of the displacement gradient matrix condenses various strain effects into a single, meaningful metric that describes the volumetric strain at a point.
Key Concepts
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Local Volumetric Strain: It defines the change in volume per unit volume at a given point during deformation.
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Scalar Triple Product: A mathematical expression involving three vectors that computes the volume of the parallelepiped.
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Determinant Relation: Volumetric strain is tied to the determinant of the deformation gradient matrix.
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Trace of Displacement Gradient: The volumetric strain can be expressed as the trace of the displacement gradient matrix.
Examples & Applications
If a cube of side length 2 meters is compressed to a cube of side length 1.8 meters, its volumetric strain can be calculated to reflect the change in volume.
In a material that undergoes uniaxial compression, the change in volume can be explored through volumetric strain metrics involved in the deformation process.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
For volume changes, don't you see, volumetric strain is the key!
Stories
Imagine a sponge being squeezed; its volume changes dramatically just like materials under stress. This shows how volumetric strain gives us direct insight into deformation.
Memory Tools
To remember volumetric strain: 'Volume's Change Is the Game'.
Acronyms
VSV
Volume Strain Variation.
Flash Cards
Glossary
- Volumetric Strain
It is the change in volume per unit volume of a local element during deformation.
- Scalar Triple Product
A mathematical operation on three vectors that gives a scalar value equal to the volume of the parallelepiped formed by them.
- Deformation Gradient
A tensor that measures the change in position of a material point due to deformation.
- Determinant
A scalar value that can be computed from the elements of a square matrix, providing critical information about the matrix.
- Trace of a Matrix
The sum of the diagonal elements of a square matrix.
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