Understanding the Principles of Hashin-Shtrikman Bounds: A Simple Explanation

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The Hashin-Shtrikman bounds provide a framework for estimating the effective properties of composite materials, particularly focusing on the conditions under which certain tensors are positive or negative definite. A user is seeking assistance in deriving the Hashin-Shtrikman formula for the permittivity of a homogeneous uniaxially anisotropic material, having already obtained the formula but lacking the derivation process. The discussion emphasizes the need for a clear and simple explanation of the principles behind these bounds. Understanding the functional approach and the significance of tensor selection is crucial for applying these concepts effectively. Overall, the thread highlights a collaborative effort to clarify complex material property estimations.
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Is anyone familiar with the Hashin-Shtrikman bounds and the principles behind them? Will you be so kind as to post a simple explanation?

So far, what I have grasped is that they came up with a functional and the bounds are obtained when a certain tensor is chosen to be positive definite/negative definite.
 
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hi
I am working on the same problem Hashin-Shtrikman formula for permittivity of a homogeneous uniaxially anisotropic material.I have the formula of permittivity of the same material.

But I don't know to derive it . Please help me If you have got any idea in this topic.
 
handsomecat said:
Is anyone familiar with the Hashin-Shtrikman bounds and the principles behind them? Will you be so kind as to post a simple explanation?

So far, what I have grasped is that they came up with a functional and the bounds are obtained when a certain tensor is chosen to be positive definite/negative definite.

hi
I am working on the same problem Hashin-Shtrikman formula for permittivity of a homogeneous uniaxially anisotropic material.I have the formula of permittivity of the same material.

But I don't know to derive it . Please help me If you have got any idea in this topic.
 
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