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## Maxwell materialA ## Additional recommended knowledge
## DefinitionThe Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected consecutively, as shown in the diagram. In this configuration, under an applied axial stress, the total stress, σ - σ
_{Total}= σ_{D}= σ_{S} - ε
_{Total}= ε_{D}+ ε_{S}
where the subscript D indicates the stress/strain in the damper and the subscript S indicates the stress/strain in the spring. Taking the derivative of strain with respect to time, we obtain: where E is the elastic modulus and η is the material coefficient of viscosity. This model describes the damper as a Newtonian fluid and models the spring with Hooke's law. If we connect these two elements in parallel, we get a model of Kelvin-Voigt material. In a Maxwell material, stress σ, strain ε and their rates of change with respect to time or, in dot notation: The equation can be applied either to the shear stress or to the uniform tension in a material. In the former case, the viscosity corresponds to that for a Newtonian fluid. In the latter case, it has a slightly different meaning relating stress and rate of strain. The model is usually applied to the case of small deformations. For the large deformations we should include some geometrical non-linearity. For the simplest way of generalizing the Maxwell model, refer to the Upper Convected Maxwell Model. ## Effect of a sudden deformationIf a Maxwell material is suddenly deformed to a strain of ε The picture shows dependence of dimensionless stress
upon dimensionless time λ If we free the material at time Since the viscous element would stay where it is, the irreversible component of deformation can be simplified to the expression below: ## Effect of a sudden stressIf a Maxwell material is suddenly subjected to a stress σ If at some time The Maxwell Model is not ideal for predicting the creep behavior of a material since it describes the strain relationship with time as linear. If a small stress is applied for a sufficiently long time, then the irreversible stresses become large. Thus, Maxwell material is a type of liquid. ## Dynamic modulusThe complex dynamic modulus of Thus, the components of the dynamic modulus are : and The picture shows relaxational spectrum for Maxwell material.
## References- http://stellar.mit.edu/S/course/3/fa06/3.032/index.html
## See alsoCategories: Non-Newtonian fluids | Materials science |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Maxwell_material". A list of authors is available in Wikipedia. |