Tensor, deformation velocity strain

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

The arithmetic mean of both angular velocities is called the strain tensor, often also called the deformation velocity tensor... 

Pressure Velocity Strain Viscosity Density Acceleration Gradients Tensor Volume Flow Force Melting Deformation Mixing Residence Time Distribution... 

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

Now that we have discussed the geometric interpretation of the rate of strain tensor, we can proceed with a somewhat more formal mathematical presentation. We noted earlier that the (deviatoric) stress tensor t related to the flow and deformation of the fluid. The kinematic quantity that expresses fluid flow is the velocity gradient. Velocity is a vector and in a general flow field each of its three components can change in any of the three... 

Here, and 2 the first and second stress difference coefficient functions, and the derivative of the strain rate is the Jaumann derivative, which is related to a frame of reference that translates and rotates with the local velocity of the fluid (this relationship can be numerically evaluated from the deformation and vorticity tensors). 

A step shear deformation A can be regarded as equivalent to a constant strain rate X/5t being applied over a very short time interval St (cf. Eq. (4.27)). The velocity gradient tensor K = (VV) is expressed by... 

The main difference between constitutive relations for simple and complex fluids lies in the fact that simple fluids satisfy the linear relationships between the stress tensor (r, t) and the strain or deformation tensor 6(r, f) or the velocity gradient Vv. In contrast, complex fluids have to incorporate a relaxation term for the stresses and a non-linear dependence on the strain or velocity gradient tensors. 

Here p is the density, t the time, Xi the three Cartesian coordinates, and o,- the components of velocity in the respective directions of these coordinates. In equation 2, the index j may assume successively the values 1, 2, 3 gj is the component of gravitational acceleration in the j direction, and atj the appropriate component of the stress tensor (see below). (A third equation, describing the law of conservation of energy, can be omitted for a process at constant temperature the discussion in this chapter is limited to isothermal conditions.) Now, many experiments are purposely designed so that both sides of equation 1 are zero, and so that in equation 2 the inertial and gravitational forces represented by the first and last terms are negligible. In this case, the internal states of stress and strain can be calculated from observable quantities by the constitutive equation alone. For infinitesimal deformations, the appropriate relations for viscoelastic materials involve the same geometrical form factors as in the classical theory of equilibrium elasticity they are described in connection with experimental methods in Chapters 5-8 and are summarized in Appendix C. 

In defining the material functions that describe responses to simple-shear deformations, a standard frame of reference has been adopted. This is shown in Fig. 10.4. The shear stress <7is the component < i (equal to <7i2 because of the symmetry of the stress tensor), and the three normal stresses are <7u, in the direction of flow (xj), Gjj in the direction of the gradient and <733, in the neutral (x ) direction. As this is by definition a two-dimensional flow, there is no velocity and no velocity gradient in the Xj direction. However, in describing shear flow behavior, we will follow the conventional practice of referring to the shear stress as <7, and the shear strain as y, where neither symbol is in bold or has subscripts. 

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