Deformation gradient tensor, components

Here we have introduced a tensor F known as the deformation gradient tensor whose components reflect the various gradients in the deformation mapping and are given by... 

The matrix F and its components are referred to as deformation gradients, F is also termed the deformation gradient tensor. It defines a transformation of the undeformed state onto the deformed state. 

The following examples illustrate how the deformation gradient tensor works. Table 1.4.1 gives the components of F in rectangular, cylindrical, and spherical coordinates. 

The deformation applied is characterized by the deformation-gradient tensor, F, with components (for a rectangular coordinate system) equal to... 

The nonzero components of the transpose of the deformation-gradient tensor are... 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

Tensor whose components are deformation gradients in an elastic solid. 

The components of the relative deformation gradient Fy and the symmetrical tensor Jy can be calculated, respectively, from Eqs. (13.7) and (13.8), giving... 

For evaluation of DUEVs, with less shear viscosity contribution to the total response, the modified stagnation technique is the procedure of preference. For practical applications the vacuum-suction filament-draw technique is probably the more valuable because the deformation rates in many applications are not solely extensional in nature. The comments regarding the velocity gradients in the tubeless siphon (41) are appropriate to the fiber-suction approach flow in a tubeless siphon approximates extensional flow in a sense that the largest components of the velocity gradient tensors are diagonal ones. ... 

Exactly the same technique can be used to analyse the Cauchy-Green tensor C. When a deformation gradient F includes rigid-body rotation, it is necessary to first form the Cauchy-Green measure C and then find its principal components and directions using the methods outlined above for V. The principal directions of C are the same as those of the pure deformation V that underlies F (F = VR). Writing the analogue of Equation (3.12) forC... 

Like the stress tensor, the deformation gradient has up to nine components, each with a scalar magnitude dxildxj and two directions for each of them. One direction comes from the unit vectors of the coordinate system used to describe x and the other from the x unit vectors. And like the stress tensor, F is a machine, a mathematical operator. It transforms little material displacement vectors from their past to present state, faithfully following the material deformation. 

The first component of the angmented Hybrid Model relates to the elastic behavior of UHMWPE, which is captured by the elastic spring (E) in the rheological representation and the elastic deformation gradient F. The polar decomposition [26] of F involves a left stretch tensor, V , and a rotation tensor, R . Using V , the logarithmic tme strain (E )... 

To describe deformation, we will examine the relative displacement of two neighboring fluid particles. At time tg, these particles are separated by the vector dx(fo) and at time by the vector dx(ti). A quantity that provides complete information about the relative displacement of any two such particles in a very small volume of the fluid is the displacement gradient tensor F whose components are given by ... 

Example 10.4 Obtain expressions for the deformation gradient and Cauchy tensors for the shear deformation illustrated in Figure 10.9. Here, the only nonzero velocity component is Uj, and it equals yx2, where y is the constant shear rate. 

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

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... 

As an example, we shall consider simple shear when z/12 0, and find components of the tensor of the recoverable displacement gradients A12, An, A22, A33 the components of the tensor are calculated from the relaxation equations (9.49) or (9.58). In this case the matrix of the deformation tensor is determined as follows... 

In the case of a simple shear deformation, schematically indicated in Figure 4.6b, the only nonzero components of the displacement gradient and strain tensors are given by... 

The possible development of gradients in the components of the interfacial stress tensor due to flow of an adjacent fluid implies that the momentum flux caused by the the flow of liquid at one side of the interface does not have to be completely transported across the interface to the second fluid but may (partly or completely) be compensated in the interface. The extent to which this is possible depends on the rheological properties of the interface. For small shear stresses the interface may behave elastically or viscoelastically. For an elastic interfacial layer the structure remains coherent the layer will only deform, while for a viscoelastic one it may or may not start to flow. The latter case has been observed for elastic networks (e.g. for proteins) that remciln intact, but inside the meshes of which liquid can flow leading to energy dissipation. At large stresses the structure may yield or fracture (collapse), leading to an increased flow. 

We can proceed the same way as with the Newtonian fluid case up to the point where we need to express the stress components in terms of the velocity gradients. Next we need the magnitude of the rate of deformation tensor... 

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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