Strain, tensor

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)  

The practical significance of Eqs. (2.28) and (2.29) lies in that, because of the positive definiteness of the symmetric tensor C, once F is known one can determine 

Let us now consider steady-state simple shear flow, for which we have the velocity field of the form 

Therefore, the covariant components of the finite strain tensor are 

We have shown here that the Cauchy-Green and Finger tensors are not equivalent measures of finite strain, which is a very important fact to remember in the formulation of constitutive equations, as is discussed in Chapter 3. 

The assembly of values of forms a second-order tensor associated with the coordinates of the point A. This tensor has the components 

It can easily be shown that Cy is a symmetrical tensor, that is ey = ep. For very small deformations, the partial derivatives dujdxj in Eq. (4.21) can be considered infinitesimal quantities of first order, so the components of the ey tensor can be approximately expressed as 

For small displacements, tan 0 0. However, if the force acts along the X2 

It is obvious that jy = jp and (Oy = —co/f. In other words, jy and (Oy are, respectively, symmetrical and antisymmetrical tensors. The strain tensor can 

In a viscoelastic as in a perfectly elastic body, the state of deformation at a given point is specified by a strain tensor which represents the relative changes in dimensions and angles of a small cubical element cut out at that position. The rate of strain tensor expresses the time derivatives of these relative dimensions and angles. Similarly, the state of stress is specified by a stress tensor which represents the forces acting on different faces of the cubical element from different directions. For details, the reader is referred to standard treatises.  

For an inflnitesiinal deformation, the components of the inBrntesimal strain tensor in rectangular coordinates with the three Cartesian directions denoted by the subscripts 1,2,3 are 

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. 

The above solid is anchored at the origin so that it cannot translate. The deformation produced by the stress tensor can be represented by tj, which are the dimensional changes along each of the respective coordinate axes. The tensile strains are the change in length per unit length or exx = d /dx, Cyy = dr ldy, e z = dljdz. 

Stresses acting on the faces on a stationary elastic cube anchored at the origin. 

Thus we can also write the strain as a symmetric tensor of rank 2, 

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. 

Since F = F, then B = C = F-F = F and it does not matter which deformation measure is used. 

Here we see that B and C do have different components for a shear deformation. Note that both tensors are symmetric, as they must be. 

Cij also gives the same result, just the identity tensor I. 

The last result in Example 1.4.2 says that there is no area or length change in the sample for the solid body rotation (i.e., there is no deformation). This is what we expect deformation tensors should not respond to rotation. They are useful candidates for constitutive equations, for predicting stress firom deformation. 

In this Eq. (1.48), the Ax and Au in Fig. 1.25 are replaced by dx and du or in terms of their partials as dx and du, thus indicating the instantaneous change of u with respect to x. Recall that ei = en = e = e x, depending on the type of notation. Nomenclatures may vary in accordance with the original research applications. For instance, the Voigt notation is useftil in calculations involving constitutive models, such as the generalized Hooke s Law, as well as for finite element analysis. 

To show more details of the angular deformation shown in Fig. 1.27d, it has been redrawn and illustrated in Fig. 1.28. 

one can state that arbitrary shear strain is the result of pure shear and some rotation occurring simultaneously during a deformation process. 

As seen in Fig. 1.27c or 1.28, strains are associated with displacements. In the general, three-dimensional case, the displacement of a point in an elemental body is denoted by u, v, and w, which are functions of position in the coordinate system X, y and z. In other words  

Returning to the two-dimensional case indicated for the shear strain in Fig. 1.30, the deformation may be expressed in terms of the displacements. In Fig. 1.26, the normal strain components were indicated in terms of u and v, which are displacement components in the x, y directions. (Other symbols are also used, such as u, V and w in the three-dimensional case.) In fact, displacement may be described in terms of its components by projecting the displacement vector onto the coordinate axes. In this manner, the average, normal-strain components, in terms of u and v after deformation, may be given as in Eq. (1.50), and in the limit, the displacement vector as in Eq. (1.50a)  

---

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

The right Cauchy-Green strain tensor corresponding to this deformation gradient is thus expressed as... 

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. 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

These are linear equations which give the symmetry of the strain tensor Sij = Sji- In the general case, the strain tensor is nonlinear,... 

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

The stress and strain tensors aij u),Sij u) are defined by the Hooke and Cauchy laws... 

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,... 

We assume that the physical parameters of the lower plate coincide with those of the upper plate in particular, the stress tensors and strain tensors of the lower plate satisfy (3.44). The thickness of the lower plate is 2s. The following conditions are considered at the external boundary T ... 

The symbols Sij = SijiyV) stand for the components of the strain tensor of the mid-plane of the plate ... 

Here Sij u) = uij + Uj,i)/2 are the components of the strain tensor. We consider function spaces whose elements are characterized by the conditions... 

Here i —> i is a continuous convex function describing the plastic yield condition. The equations (5.7) provide a decomposition of the strain tensor Sij u) into a sum of an elastic part aijuicru and a plastic part ij, and (5.6) are the equilibrium equations. 

The functions v,aij,Sij v) represent the velocity, components of the stress tensor and components of the rate strain tensor. The dot denotes the derivative with respect to t. The convex and continuous function describes the plasticity yield condition. It is assumed that the set... 

The strain tensor 5 can be written for noncentrosymmetry point group crystals as ... 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

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. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

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... 

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. 

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. 

The referential constitutive equations for an inelastic material may be set into spatial terms. Casey and Naghdi [14] did so for their special case of finite deformation rigid plasticity discussed by Casey [15], Using the spatial (Almansi) strain tensor e and the relationships of the Appendix, it is possible to do so for the full inelastic referential constitutive equations of Section 5.4.2. 

It is possible to use directly as a measure of the irrotational part of the deformation, but it is more convenient to use the strain tensor... 

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 ... 

---