Deformation gradient defined

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 ... 

In such cases W will be a function of position as well as of temperature and the coordinates of the deformation gradient tensor. Finally, most materials, in particular polymers, are anelastic. Energy is dissipated in them during a deformation and the stored energy function W cannot be defined. It is still of value, however, to consider ideal materials in which W does exist and to seek its form since such ideal materials may approximate quite closely to the real ones. 

In Chapter 5, we defined the deformation gradient tensor E, Cauchy tensor C, and Finger tensor B, respectively, as... 

In continuum mechanics, we say that dX is mapped into dx and the mapping is defined as the deformation gradient ... 

As shown in Figure 4.2, a material body can be first stretched and then rotated to the deformed shape it can also be rotated first and then stretched to the deformed shape. This type of decomposition is similar to the decomposition of motion in dynamics. In Figure 4.2, we also highlight the change of a differential element with stretch and rotation. Let us define the rotation deformation gradient as R and the stretch deformation gradient as U (stretch first) or V (rotation first). 

We have known that the deformation gradient, F, is defined in the undeformed body, that is, the Lagrangian quantity. We now want to know the relation between L and F so that conversion between the Lagrangian and Eulerian quantities becomes possible. The rate of deformation gradient is... 

Note that both N and n are unit tangent vectors to the area elements dA and da, respectively. In Eqs. (8.5) and (8.6), F denotes the deformation gradient tensor to be calculated from the fluid element trajectories through integration of Eq. (8.3). The ratio of intermaterial areas, also called area stretch, r) can be defined as ... 

Note 2.3 (Small strain theory). In most textbooks on elasticity theory a displacement vector is defined as u = x — X. However we know that x=XiCi, X = XiEi and the transformation of both bases are locally defined by the deformation gradient F therefore it is difficult to introduce the globally defined displacement vector u unless a common rectangular Cartesian coordinate system is used. 

Recall the definition of the deformation gradient dx = F dX its material time-derivative defines the following velocity gradient ... 

If an orthonormal coordinate transformation tensor Q (e- = eo is used instead of the deformation gradient F, the concept of the convected derivative can be extended. That is, let ft = 2 2 an antisymmetric rotation tensor generated by Q as shown in (2.27), then the corotational derivative of a second-order tensor T due to Q is defined by... 

Since the deformation gradient F can be written in terms of the polar decomposition as defined in (2.46), its time-differentiation gives... 

By introducing the reference configuration for the entire solid phase, we define the basis Ek for the reference configuration, and the reference point is given by X = X Ek- For the current configuration the basis is e,, and the current point of the solid phase is given by jc = x, e, (which is the same as the current point of the fluid phase). A deformation gradient of the solid phase is f. Then we have... 

In the form defined in this section, the true strain (p can only be used for normal strains, but it cannot describe large shear deformations. To describe large and arbitrary deformations in more than one dimension, several approaches can be used [16,67,80]. They are all based on a matrix, called deformation gradient F. The deformation gradient is similar to a coordinate transformation from the undeformed to the deformed state. 

Polymer melts rarely behave as Newtonian fluids, but we can still determine the viscosity with this experiment. Let us suppose that the stress depends on the deformation gradient in a completely arbitrary way. With the assumption we have made about the flow, the only nonvanishing part of the deformation gradient, regardless of how we define it, is dv ldy. Thus, while Xyx may be an arbitrary function of dvx/dy, the condition Xyx = constant means that dvx/dy must also be constant. Thus, Equations 2.36a-b still follow, and the definition of the shear rate is unchanged. 

The constraint (53) defines an hypersiuface in the space of the deformation gradients. Any stress normal to this surface, i.e., in the directiOTi 3detC/9F, does not expend work on any (virtual) incremental deformaticMi 8x compatible with the constraint. The stress is, hence, determined by the constitutive law unless a vector parallel to 3detC/0F. From an energetic point of view this is tantamount to assume the strain energy function as... 

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 deformation gradient F contains geometrical information that is not related to strain (i.e. dimensional or shape change) namely, it includes rigid-body rotations. The matrix associated with rotation is familiar from its use in changing the representation of a vector when the axis set is rotated. It also defines the rigid-body rotation of an element of material. 

The eigenvectors of C define the direction cosines of the I and n directions with respect to the original 1- 2 axes. They give the angle of transformation 6 and the associated rotation matrix Re required to transform V back to the 1-2 axes set to give the deformation gradient V ... 

Now consider two neighboring material points X, and X,+dX, which were located at neighboring places X z) and Jr(r)-l-dAr(r) at time r. The relative deformation gradient tensor F is defined by... 

Further, the stress-strain behavior is defined using the following equations. Writing the current position of a material point as x and the reference position of the same point as X, the deformation gradient is then defined as... 

For simplicity, the deformation gradient with the volume change eliminated is defined as... 

Using the inverse of the deformation gradient we can also define inverses of the deformation tensors B and C... 

We can define other deformation tensors, also, in terms of the deformation gradient tensor F. According to the polar decomposition theorem of the second-order tensor (see Appendix 2A), the deformation gradient tensor F, which is an asymmetric tensor and is assumed to be nonsingular (i.e., det F 0), can be expressed as a product of a positive symmetric tensor with an orthogonal tensor (Jaunzemis 1967) ... 

In the field of mathematics, the requirements for linearity are the same whether they be applied to differential equations, functions, operators, transforms, functionals or other mathematical operations. When these linearity requirements are applied to constitutive theories they are applied best to functional equations [21] since in continuum mechanics a simple material is defined as material wherein the present state of stress can depend upon the history of the deformation gradients [18,19,20]. For the case of small displacements and rotations, the state of stress can be taken as being dependent upon the history of the strain tensor [21-26] and can be expressed functionally as... 

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