Small-strain tensor

To provide an elementary treatment, in this seetion the theory is eon-strueted in terms of the elassieal small strain tensor s defined as the symmetrie... 

The theory of Section 5.2 was developed using the classical small strain tensor E, implicitly assuming that deformations are small in the sense of Section A.7. If deformations are indeed small, then the approximations in Section A.7 hold. In particular, from (A.IOO2) and (A.103), neglecting higher-order terms. 

Here (eij) = (4 (x w)jis the matrix of dielectric moduli in the solid phase, e(u) stands for the small strain tensor, and... 

We next introduce the small strain tensor ey and its deviatoric part... 

As we will see in subsequent chapters, for many purposes (e.g. the linear theory of elasticity) the small-strain tensor suffices to characterize the deformation of the medium. 

Since the third term of the rh.s. is second-order infinitesimal, the small strain tensor is given by... 

In these relations angular brackets denote volume averages of the Cauchy stress tensor and the small strain tensor, respectively. For the correct use of volmne and ensemble averages in connection with random and periodic microstructures the reader should consult [Torquato... 

Integral-type constitutive equations may also be written in terms of strain tensor (or relative strain tensor), instead of rate-of-strain tensor. For infinitesimally small deformations, if the response of a system can be expressed by the linear superposition of a series of separate responses at different times to a series of step changes in the input, the stress tensor a as a response can be expressed in terms of the infinitesimally small strain tensor e by... 

In what follows the Kirchhoff-Love model of the shell is used. We identify the mid-surface with the domain in R . However, the curvatures of the shell are assumed to be small but nonzero. For such a configuration, following (Vol mir, 1972), we introduce the components of the strain tensor for the mid-surface,... 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

We can note that the change of volume during the deformation of a body is usually small, so we can assume further that the original strain tensor can be used instead of the newly introduced one. 

Stress-Strain Relations as Equations of State. Simple theory of elasticity assumes that the material is isotropic and that induced stresses and strains are linearly related to each other as long as they are small. The theory further assumes that the stress and the strain tensors always have the same axes. Poisson s ratio and... 

Here we have used the fact that Fij = 5 + Uij. In addition, we have invoked the summation convention in which all repeated indices (in this case the index k) are summed over. For the case in which all the displacement gradient components satisfy Uij 1, the final term in the expression above may be neglected, resulting in the identification of the small strain (or infinitesimal strain) tensor,... 

To evaluate the integrals in (5 49), we would need to determine the components of the rate-of-strain tensor in a Cartesian coordinate system with axes in the horizontal and vertical directions from the velocity components u -) and u(q, given by (5 40a) and (5-40b). This is not a difficult task. However, in the present case, we will be content to show that F-v) is 0(e ]) and is thus asymptotically small compared with l< (yp). To see this, we note that... 

Problem 7-9. Motion of a Force- and Torque-Free Axisymmetric Particle in a General Linear Flow. We consider a force- and torque-free axisymmetric particle whose geometry can be characterized by a single vector d immersed in a general linear flow, which takes the form far from the particle y°°(r) = U00 + r A fl00 + r E00, where U°°, il, and Ex are constants. Note that E00 is the symmetric rate-of-strain tensor and il is the vorticity vector, both defined in terms of the undisturbed flow. The Reynolds number for the particle motion is small so that the creeping-motion approximation can be applied. 

While we do not want to give a sophisticated model including all the effects found in the mechanical behavior of polymers, we restrict ourselves to the simplest case, namely to an elastic small-strain model at constant temperature. Therefore, the governing variables are the linear strain tensor [Eq. (13)] derived from the spatial gradient of the displacement field u, and the microstructural parameter k and its gradient. The free energy density is assumed to be a function of the form of Eq. (14). 

Tensors for the small-strain elastic properties of materials... 

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