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I was looking at a paper on a viscoelastic material and the author derived a formula that featured shear modulus and shear viscosity. I don't really see what the difference between these two parameters is, they both seem to mean the same thing as far as I can see, which is resistance to shearing of the material? So what differentiates them?

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Shear viscosity is relevant when there is a velocity gradient in the material. The shear modulus applies in the static case.

For viscoelastic materials, both factors may matter. In principle, if you suddenly apply a shear to a material, there will be initial resistance to the motion due to the viscosity: once the molecules settle, only the elasticity provides a restoring force.

What differentiates them, then, is this: shear modulus refers to static phenomena, shear viscosity refers to dynamic phenomena.

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    $\begingroup$ As a person trained in rheology, I would say the opposite of this. If a shear stress is suddenly applied to a viscoelastic fluid, the elastic behavior of the fluid is emphasized, and there will be an instantaneous shear strain, equal to the shear stress divided by the shear modulus. If the shear stress is maintained for a long period of time, the viscous behavior of the fluid is emphasized, and the rate of strain (i.e., the shear rate) will be equal to the shear stress divided by the shear viscosity. Rapid deformations favor elastic behavior; slow persistent deformations induce viscous. $\endgroup$ Nov 15, 2017 at 20:29

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