Understanding Strain Invariants in Strength of Materials

In summary, stress and strain invariants are mathematical functions which describe the behaviour of materials without depending on the orientation of the axes used to describe them.
  • #1
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Can anyone help me understand the concept of a "strain invariant" given a strain state matrix? or perhaps point me towards something?

thanks for any help.
 
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  • #2
This is what the "big idea" of stress and strain invariants is about. The details of the maths is fairly easy to find with Google.

If you have a structure, the way it behaves (the stresses, deflections, failure mode, etc) doesn't depend on where you choose to put the X Y Z axes when you set up model and solve the equations.

But the numbers representing the 6 components of stress and strain at a point (3 direct and 3 shear) DO depend on what direction the axes point in. So, those raw numbers are not always the right things to use, to understand what the solution means.

It turns out there are some functions of the 6 stress/strain components which are independent of the orientation of the axes. There are three of these functions which are called the stress/strain invariants.

If you have an isotropic material, then the physics of the material behaviour is also independent of the orientation of the axes used to describe it. Therefore, the material behaviour (elasticity, plasticity, etc) can be described using the 3 invarants. In fact you must be able to write these formulas using only the invariants, otherwise the material behaviour would not be isotropic.

The first invariant (in the standard numbering convention) is the sum of the direct stresses or strains e_xx + e_yy + e+zz. Physically, that represents the "fluid presssure component" part of a the stress at the point, or the "uniform expansion or compression" part of the strain.

The second invariant is more complicated mathematically, but for stress it's very similar to the Von Mises stress function. It's a measure of the "average amount of shear" at the point. This is why Von Mises stress is a good failure measure for ductile materials which fail because of shear stress not direct stress. The actual formula for VM stress doesn't look much like a "shear stress" at first sight, but if you think about a shear stress, then rotate the axes 45 degrees so the shear is equivalent to tensile and compressive direct stresses, the connection between the VM "difference of principal stresses, squared" formula appears.

The third invariant isn't so simple to understand physically, but the maths says there are three of them, so it is what it is.

Hope that helps get started.
 
  • #3


Sure, I would be happy to help you understand the concept of a "strain invariant" in Strength of Materials. A strain invariant is a physical quantity that remains constant regardless of the coordinate system or orientation of the material. In simpler terms, it is a property of a material that does not change under different loading conditions.

To understand this concept, let's first define what strain is. Strain in Strength of Materials refers to the deformation or change in shape of a material when subjected to external forces. It is typically represented by a strain state matrix, which is a mathematical representation of the strain in all directions.

Now, a strain invariant is a property of a material that can be calculated using the components of the strain state matrix. It is a scalar quantity, meaning it has only magnitude and no direction. The most commonly used strain invariants are the first and second invariants, also known as the principal strains.

The first invariant, also denoted as I1, is the sum of the three principal strains and represents the total strain energy in a material. It remains constant under any loading condition, making it a useful measure of the overall deformation of a material.

The second invariant, also denoted as I2, is the sum of the products of the principal strains taken two at a time. It is related to the distortion or shear deformation in a material and is also a constant value for a given material.

Understanding strain invariants is crucial in Strength of Materials because they provide a more comprehensive understanding of the deformation and behavior of materials under different loading conditions. By calculating and analyzing these invariants, engineers can make informed decisions about material selection, design, and structural integrity.

I hope this explanation helps you understand the concept of a strain invariant and its significance in Strength of Materials. If you need further clarification or would like to explore the topic in more detail, I suggest consulting your textbook or referring to online resources such as lecture notes or video tutorials. Best of luck!
 

What are strain invariants in the context of strength of materials?

Strain invariants are mathematical expressions that help quantify the deformation or strain experienced by a material when subjected to external forces. They are used to understand the behavior of materials under different loading conditions and can provide insight into their strength and stability.

How are strain invariants calculated?

There are three commonly used strain invariants: the first, second, and third invariants. The first invariant is the trace of the strain tensor, the second invariant is the square root of the second deviatoric strain invariant, and the third invariant is the determinant of the strain tensor. These invariants can be calculated using the strain components in different directions.

What is the significance of strain invariants in strength of materials?

Strain invariants are important because they provide a way to describe the deformation of a material in a way that is independent of the coordinate system. This allows for a more accurate and consistent analysis of a material's behavior under different loading conditions.

How do strain invariants relate to stress invariants?

Stress invariants are related to strain invariants through the material's elastic properties. The first stress invariant is proportional to the first strain invariant, while the second and third stress invariants are proportional to the second and third strain invariants, respectively.

What are some practical applications of understanding strain invariants in strength of materials?

Strain invariants are useful in various engineering applications, such as designing structures and analyzing material failure. They can also provide valuable information for material selection and optimization, as well as for predicting the behavior of materials under different loading conditions.

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