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Upper Convected Maxwell model : ウィキペディア英語版
Upper-convected Maxwell model
The upper-convected Maxwell (UCM) model is a generalisation of the Maxwell material for the case of large deformations using the upper-convected time derivative. The model was proposed by James G. Oldroyd. The concept is named after James Clerk Maxwell.
The model can be written as:
: \mathbf + \lambda \stackrel
where:
* \mathbf is the stress tensor;
* \lambda is the relaxation time;
* \stackrel \mathbf + \mathbf \cdot \nabla \mathbf -( (\nabla \mathbf)^T \cdot \mathbf + \mathbf \cdot (\nabla \mathbf))
*\mathbf is the fluid velocity
*\eta_0 is material viscosity at steady simple shear;
*\mathbf is the tensor of the deformation rate.
==Case of the steady shear==
For this case only two components of the shear stress became non-zero:
:T_=\eta_0 \dot \gamma \,
and
:T_=2 \eta_0 \lambda ^2 \,
where \dot \gamma is the shear rate.
Thus, the upper-convected Maxwell model predicts for the simple shear that shear stress to be proportional to the shear rate and the first difference of normal stresses (T_-T_) is proportional to the square of the shear rate, the second difference of normal stresses (T_-T_) is always zero. In other words, UCM predicts appearance of the first difference of normal stresses but does not predict non-Newtonian behavior of the shear viscosity nor the second difference of the normal stresses.
Usually quadratic behavior of the first difference of normal stresses and no second difference of the normal stresses is a realistic behavior of polymer melts at moderated shear rates, but constant viscosity is unrealistic and limits usability of the model.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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