Pure Shear Questionthanks for your help

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In summary, the conversation is about a person struggling to prove that a state of pure shear does not cause any volume change in linear elastic deformation. They have a good understanding of shear stresses and strains but are having difficulty with this specific problem. They have tried using strain and stress equations and Young's modulus, but are still struggling. They also ask for help in understanding the expression for volume deformation and the values of principal stresses in a pure shear state.
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Homework Statement


Hey everyone. I was practicing some problems, and I came across one that seems pretty simple, yet I am having difficulty proving the answer. The question asks us to show that a state of pure shear causes no volume change in linear elastic deformation. I have a very good understanding on shear stresses and strains, however, this problem seems harder then it actually is. If someone could point me in the right direction I would appreciate it greatly!


Homework Equations


Strain Eq, Stress Eq, Young's Modulus (standard elementary equations)


The Attempt at a Solution



I have literally spent hours reading my book and working out problems, such as these. I seem to understand all of the formulas, however, I am having trouble proving the statement.
 
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What is the expression for volume deformation, i.e. how does it relate with the principal stresses? What are the values of these stresses for the pure shear state?
 

1. What is pure shear?

Pure shear is a type of deformation that occurs when a material is subjected to two equal and opposite forces that act in parallel to each other, causing the material to distort without changing its volume.

2. How is pure shear different from simple shear?

In simple shear, the forces act in opposite directions but not necessarily parallel, resulting in a change in both shape and volume of the material. In pure shear, the forces act in parallel, causing only a change in shape.

3. What are some examples of pure shear in nature?

Some examples of pure shear in nature include the folding of rocks in mountains, the stretching and thinning of ice sheets, and the deformation of soft tissues in the human body.

4. How is pure shear studied in the laboratory?

In the laboratory, pure shear can be studied using specialized equipment such as a shear box or a torsion machine. These machines apply equal and opposite forces in parallel to a material, allowing researchers to observe its behavior under pure shear.

5. What are the practical applications of studying pure shear?

Studying pure shear can help scientists and engineers better understand the mechanical properties of materials and how they respond to external forces. This knowledge can be applied in various industries, such as construction, manufacturing, and geology, to design and improve structures and products.

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