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Finger tensor defined

Another combination of the displacement gradient tensors which are often used are the Cauchy strain tensor and the Finger strain tensor defined by B —1 = Afc A and B = EEt, respectively. [Pg.82]

In Chapter 5, we defined the deformation gradient tensor E, Cauchy tensor C, and Finger tensor B, respectively, as... [Pg.113]

Here B is the standard Finger strain tensor used in continuum mechamcs, and Yfo] = 8 — B IS a relative finite strain tensor, defined in DPL, Eq. D.3-4. We note in passing that it follows from Eq. (13.5) that the quantity HfkT) QQ P, dQ is equal to oi and thus satisfies Eq. (13.6). [Pg.58]

This is the same as DPL, Eq. (15.3-17). Thus the stress tensor is given in terms of the Finger strain tensor via Eq. (13.10). The polymer contribution to the stress tensor for the Hookean dumbbell model is obtained by replacing in Eq. (14.12) by the a tensor defined m Eq. (13.7)... [Pg.66]

Thus, physically the Finger tensor describes the area change around a point on a plane whose normal is n. B can give the deformation at any point in terms of area change by operating on the normal to the area defined in the present or deformed state. Because area is a scalar, we need to operate on the vector twice. [Pg.31]

Length, area, and volume change can also be expressed in terms of the invariants of B or C (see eqs. 1.4.45-1.4.47). Note that the Cauchy tensor operates on unit vectors that are defined in the past state. In the next section we will see that the Cauchy tensor is not as useful as the Finger tensor for describing the stress response at large strain for an elastic solid. But first we illustrate each tensor in Example 1.4.2. This example is particularly important because we will use the results direcfiy in the next section with our neo-Hookean constitutive equation. [Pg.32]

The upper-convected time derivative is a time derivative in a special coordinate system whose base coordinate vectors stretch and rotate with material lines. With this definition of the upper-convected time derivative, stresses are produced only when material elements are deformed mere rotation produces no stress (see Section 1.4). Because of the way it is defined, the upper-convected time (teriva-tive of the Finger tensor is identically zero (see eqs. 2.2.3S and 1.4.13) ... [Pg.146]

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. [Pg.336]

The Doi-Edwards equation is a K-BKZ model, since the scalar functions and are derivatives of a strain energy function and depend on the first and second invariants of the Finger tensor, which are defined by Eqs. 10.8 and 10.9. While these two functions cannot be written in a closed form, Currie [13] has shown that they can be approximated by the following analytical expressions. [Pg.339]

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. [Pg.87]

L /polymer extensibility smectic-layer compressive modulus E E, Finger strain tensor B , Cauchy strain tensor yriso/r, capillary number characteristic ratio, defined by R )q — Ccotib translational diffusivity-------------------------... [Pg.635]

There is an alternative form of Finger s equation which gives us a choice and is, indeed, to be preferred when dealing with rubbers. We introduce the Finger strain tensor B, being defined as the reciprocal of the Cauchy strain tensor... [Pg.316]

Due to the unknown hydrostatic pressure, p, the individual normal stresses an are indeterminate. However, the normal stress differences are well-defined. Let us consider the difference between Gzz and Gxx- Insertion of the Finger strain tensor associated with uniaxial deformations, Eq. (7.80), in the constitutive equation (7.74) yields... [Pg.319]

In which C is the Finger deformation tensor (see Chapter 2) and m t) is the memory function defined by... [Pg.567]

Three scalar invariants of the Finger strain tensor may defined in the following way... [Pg.247]


See other pages where Finger tensor defined is mentioned: [Pg.119]    [Pg.239]    [Pg.2430]    [Pg.1472]    [Pg.119]    [Pg.239]    [Pg.2430]    [Pg.1472]    [Pg.89]    [Pg.337]    [Pg.382]    [Pg.88]    [Pg.114]    [Pg.315]    [Pg.446]    [Pg.376]    [Pg.110]    [Pg.688]   
See also in sourсe #XX -- [ Pg.25 ]




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