Deformation rate tensor

In the Cartesian coordinate system the deformation rate tensor, D, is defined as ... 

The shear stress tensor t for a Newtonian fluid is the product of the deformation rate tensor and the viscosity. 

The dissipation is the scalar product of the shear stress and the deformation rate tensors. 

Using the parameters in the above figure, the deformation rate tensor is defined as follows ... 

Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force... 

Purely viscous constitutive equations, which account for some of the nonlinearity in shear but not for any of the history dependence, are commonly used in process models when the deformation is such that the history dependence is expected to be unimportant. The stress in an incompressible, purely viscous liquid is of the form given in equation 2, but the viscosity is a function of one or more invariant measures of the strength of the deformation rate tensor, [Vy - - (Vy) ]. [An invariant of a tensor is a quantity that has the same value regardless of the coordinate system that is used. The second invariant of the deformation rate tensor, often denoted IId, is a three-dimensional generalization of 2(dy/dy), where dy/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is often taken to be a specific function-a power law, for example-of (illu). ]... 

Equations 9.8 and 9.10b are written for rectilinear flows and infinitesimal deformations. We need equations that apply to finite, three-dimensional deformations. Intuitively, one might expect simply to replace the strain rate dy/dt by the components of the symmetric deformation rate tensor (dv /dy -y dvy/dx, etc.) to obtain a three-dimensional formulation, as in Section 2.2.3, and dr/dt by the substantial derivative D/Dt of the appropriate stress components. The first substitution is correct, but intuition would lead us badly astray regarding the second. Constitutive equations must be properly invariant to changes in the frame of reference (they must satisfy the principle of material frame indifference), and the substantial derivative of a stress or deformation-rate tensor is not properly invariant. The properly invariant... 

The finite element procedures for the analysis of elastic>plastic solids at large strain have been given by Lee [9] and implemented by Chiou [11] and Chiou et al. [10]. In this work, only a few comments on the finite element procedures will be made. Equation (16), which links the Truesdell stress rate tensor and the deformation rate tensor, may be regarded as the stress-strain relation in rate fom with a being the "slope" at a particular point in stress space. However, in nonlinear finite element analysis, one has to have a stress-strain relation in incremental form which enables the increments in displacements, strains, and stresses not to be infinitesimally small. Therefore, it is proposed to adopt the following incremental stress-strain relation... 

A parameter known to influence flow-enhanced nucleation is the strain rate. It is customary to define an effective shear rate in terms of the deformation rate tensor D,... 

A common explanation for such phenomena is that the gradient of the deformation rate tensor affects the flow induced anisotropy and hence the stress. It follows obviously that filled (polymer) systems cannot be "simple" fluids since their behavior, by virtue of their true nature, violates the principle of "local action," which states that the stress in a fluid element is determined by the deformation history of that fluid element and is independent of the history of neighboring elements. This is the main reason for the... 

The viscoelastic mWM constitutive equation is practically the Maxwell model in which viscosity and relaxation time are allowed to vary with the second invariant of the deformation rate tensor in such a way that the extensional viscosity does not attain the infinite valne [5]. The constitntive equation of the mWM model is given by the following set of equations ... 

Tl(ll , III J = 1](II,(18) where, Ifc and IIId represent second and third invariant of deformation rate tensor, respectively. 

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