Strain tensor, definition

The St. Venant compatibility equations [13,14,15] follow immediately from the strain tensor definition of Equation (2), and are... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

In different reference systems the strain and stress tensors have different components, the transformation being easily derived starting from the definitions. Let us consider, for example, the sample reference system (y ) and denote by Latin letters etm and stm the components of the strain and stress tensors in this system. If the transformation of the sample reference system (y,) into the crystal reference system (x ) is given by Equation (1) then the transformations of the strain tensors are the following ... 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

The first one is the small strain definition, or Cauchy strain tensor. Equation (4.1) can be rewritten in tensor form as... 

Definition (7) has been invoked within the parentheses that extend the right hand side of equation (6). Divergence represents the trace (first invariant) of the transient total strain tensor of the porous granular structure. It signifies a change in volume of this saturated porous structure. This contrasts to the deviatoric component of the total structural strain tensor, which signifies a change in shape. 

In this limit it is possible to neglect their squares and products. By this approximation (3.21) reduces to the definition of the coordinates Sjd of the infinitesimal strain tensor, again a symmetrical second rank tensor... 

In most treatises,"- 3 the strain tensor is defined with all components smaller by a factor of 2 than inequation 3, so that 711 = dui/dxi and 721 = du2/bx + bui/bx ). However, such a definition makes discussion of shear or shear flow somewhat clumsy either a practical shear strain and practical shear rate must be introduced which are twice 721 and 721 respectively, or else a factor of 2 must be carried in the constitutive equations. Since most of the discussion in this book is concerned with shear deformations, we use the definition of equation 3 which follows Bird and his school" and Lodge. - This does cause a slight inconvenience in the discussion of compressive and tensile strain, where a practical measure of strain is subsequently introduced (Section F below). In older treatises on elasticity, strains are defined without the factor of 2 appearing in the diagonal components of equation 3, but with the other components the same. 

For large deformations or rates of deformation, definition of the strain tensor or rate of strain tensor becomes extremely complicated and there are various different alternatives. A thorough discussion is presented in Chapters 7-9 of reference 10. 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

It follows from these definitions that strain tensor [7.7] characterizes the strain of a solid material subjected to stress. For a solid, the constitutive equation is a relation between the stress tensor and the strain tensor ... 

Fig. 7.9. Notions used in the definition of the Cauchy strain tensor The material point at r in the deformed body with its neighborhood shifts on unloading to the position r. The orthogonal infinitesimal distance vectors dri and drs in the deformed state transform into the oblique pair of distance vectors dri and dr. Orthogonality is preserved for the distance vectors dra, drc oriented along the principal axes...

N. P. Krayt and L. Rothenburg, Micromechanical definition of strain tensor for granular materials. ASME Journal of Applied Mechanics 118,706,1996. 

Figure E. 1. Definition of the strain tensor for the cartesian (jc, y, z) coordinate system the arbitrary point in the soUd at r = xx + yy + zz in the reference configuration moves to F = r + Sr = (x + m)x + (y + w)y + (z + w)z under the action of forces on the solid. (In this example the top part is under compression, the bottom part is imder tension.)...

The definition of the strain tensor in terms of displacements u,., Uq, and the form of the equilibrium equations for these coordinates are given by Sokolnikoff (1956), pp. 184, for example. The sphere is taken to be centred at the origin. All specified quantities, namely the boundary functions and the temperature field, are assumed to be spherically symmetric. Body forces are neglected. The displacements have the form... 

The Cauchy strain tensor is symmetric by definition. Therefore, it can be converted into a diagonal form by an appropriate rotation of the coordinate system. We deal with these conditions as indicated in Fig. 9.9, by attaching, to each selected material point, a triple of orthogonal infinitesimal distance... 

The definition of the convected strain tensor involves the difference between two quantities associated with a given material point at different times, and it refers to the same material point in convected coordinates. Now, we must transform the quantities i ,y( , t) and V y( , t ) (also v (, t ), and v 9(, t)) in such a manner that they both refer to the same point in a coordinate system fixed in space, because physical quantities (kinematic and dynamic variables) can only be measured relative to a frame of reference fixed in space. This can be done by making use of the transformation relations between two coordinate systems. 

Note that from the definition of the rate-of-strain tensor d for shear flow, given by Eq. (2.12), we have 2d 2 = Y- Thus the factor 2 does not appear in Eq. (3.80). 

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. 

The scalar invariants of certain kinematic tensors play important roles in continuum mechanics, constitutive equations and kinetic theories. Of particular interest are the invariants of the rate of strain tensor and of the finite-strain tensors. There are many ways to define these invariants, and we give only those definitions that are used in later sections in addition, much research has been done on the definition of joint invariants of several tensors. ... 

Note that strains transform with the same transformation equation as stresses provided that the tensor definition of shear strain is used (see Table 8.3). 

The structure of the section is as follows. In Section 2.8.2 we give necessary definitions and construct a Borel measure n which describes the work of the interaction forces, i.e. for a set A c F dr, the value /a(A) characterizes the forces at the set A. The next step is a proof of smoothness of the solution provided the exterior data are regular. In particular, we prove that horizontal displacements W belong to in a neighbourhood of the crack faces. Consequently, the components of the strain and stress tensors belong to the space In this case the measure n is absolutely continuous with respect to the Lebesgue measure. This confirms the existence of a locally integrable function q called a density of the measure n such that... 

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