... | ... | @@ -23,7 +23,7 @@ In principle it is possible to analytically find a closed form for the Cauchy st |
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## Local energy calculation
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To relate intensity with local energy, we need to know the scaling of the force with particle deformation. The figure below experimentally investigates this relation for the sample circular particles. The relation shows a great fit with Hertzian contact force, $`f = k(\delta r)^{3/2}`$, as [expected in circular particles](https://books.google.com/books?hl=en&lr=&id=Do6WQlUwbpkC&oi=fnd&pg=PA1&dq=johnson+1987+contact&ots=gqgjhpg96V&sig=JMvJtJKYULMcacedO_4RA-MDL8w#v=onepage&q=johnson%201987%20contact&f=false). Combining Hertzian force relation and the linear relation of force and mean image intensity for a single particle, we can deduce that: $`\delta E = \frac{2}{5}k(\delta r)^{5/2} \sim (\delta f)^{5/3} \sim (\delta I_{avg})^{5/3}`$ in which $`E`$ is the stored energy in the particle, $`\delta r`$ is the displacement at the point of force contact, and $`I_{avg}`$ is the average intensity in a single particle, and $`\delta`$ represents the change in them. This equation establishes a relation between local image intensity and local energy in the granular medium that can be used to extract local energy information.
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To relate intensity with local energy, we need to know the scaling of the force with particle deformation. The figure below experimentally investigates this relation for the sample circular particles. The relation shows a great fit with Hertzian contact force, \(f = k(\delta r)^{3/2}\), as [expected in circular particles](https://books.google.com/books?hl=en&lr=&id=Do6WQlUwbpkC&oi=fnd&pg=PA1&dq=johnson+1987+contact&ots=gqgjhpg96V&sig=JMvJtJKYULMcacedO_4RA-MDL8w#v=onepage&q=johnson%201987%20contact&f=false). Combining Hertzian force relation and the linear relation of force and mean image intensity for a single particle, we can deduce that: \(\delta E = \frac{2}{5}k(\delta r)^{5/2} \sim (\delta f)^{5/3} \sim (\delta I_{avg})^{5/3}\) in which \(E\) is the stored energy in the particle, \(\delta r\) is the displacement at the point of force contact, and \(I_{avg}\) is the average intensity in a single particle, and \(\delta\) represents the change in them. This equation establishes a relation between local image intensity and local energy in the granular medium that can be used to extract local energy information.
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* Stress-strain relation for a single particle as measured experimentally. Blue and red data points correspond to two sizes of particles. The scaling shows a great match with the Hertzian contact prediction:
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