Strain components

Int internal strain component Elec electrostatic interaction component vdW van der Waals interaction component. 

In the differential models stress components, and their material derivatives, arc related to the rate of strain components and their material derivatives. 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

The stress—strain relationship is used in conjunction with the rules for determining the stress and strain components with respect to some angle 9 relative to the fiber direction to obtain the stress—strain relationship for a lamina loaded under plane strain conditions where the fibers are at an angle 9 to the loading axis. When the material axes and loading axes are not coincident, then coupling between shear and extension occurs and... 

This photoelastic stress analysis is a technique for the nondestructive determination of stress and strain components at any point in a stressed product by viewing a transparent plastic product. If not transparent, a plastic coating is used such as certain epoxy, polycarbonate, or acrylic plastics. This test method measures residual strains using an automated electro-optical system. 

At large plastic strains, the elastic strain component constitutes only a small fraction of the total strain and the volume of the deforming member can be taken to remain essentially unchanged so that one has... 

If b AGIdX4 = 0 and d AGIdx = 0, the strain components are linearly dependent on and gorf (Salje, 1985) ... 

Replacing the generalized strain e with strain components X4 and Xg and adding the elastic energy term in the Landau expansion results in equation 5.175. 

The piezoelectric coejficients are defined as the ratios of the strain components over a component of the applied electrical field intensity, for example,... 

There is a corresponding strain tensor (which is not presented here due to its complexity), and for each stress component there is a corresponding strain component. Hence, there are three normal strains, and e, and six shear strains, which... 

The boundary conditions require that at the free surface each component of traction should vanish. The traction can be found from Hooke s law (6.30), with the strain components obtained from (6.46) using (6.23), and the dilation from (6.47). The normal component of the traction is... 

Fig. 26. Computed minimum value of the magnetic field Hm necessary to align the saturation value of magnetization along the hard axis as a function of biaxial strain component for two values of the hole concentrations in Gao.95Mno.05As. The symbol (100) - 1001) means that the easy axis is along IOO], so that Hun is applied along [001] (Dietl et al. 2001c).

For a material strained in one direction, one may check that At/l equals the only nonzero strain component, when the z-direction and the measuring direction are chosen along the strain direction. 

For an isotropic material, the space spanned by the five remaining strain components cannot be reduced further. Consequently, there are only two magnetostriction modes. The energy density can be written down directly, in principle for either irreducible representation separately ... 

For uniaxial (hexagonal) symmetry the 6 strain components are subdivided in two (invariant) one-dimensional subsets (indicated by the superscript a, and subscripts 1 and 2 for the volume dilatation and the axial deformation, respectively), and two different two-dimensional subsets, indicated by y for deformations in the (hexagonal) plane, and by e for skew deformations. These modes are also depicted in fig. 3. In this case, the magnetostriction can be expressed as... 

Unlike the two-dimensional z-r and r-6 planes, where there are many practical problems are posed and solved, it is hard to think of a two-dimensional problem that is posed on a 6-z surface. Nevertheless, the strain components are certainly required in three-dimensional problems. The derivation follows the same procedure as we have just followed. However, we will not work through the derivations here but simply state the important results. 

Describing the changes in the relative arrangement of parts of the crystal before and after strain, we obtain the strain components. If x, y, z are the coordinates of a point before strain and x, y, z the coordinates after the strain, then the displacements of the point along the three coordinate axes are given by the equations... 

For each stress component a there exists a corresponding strain component y. Even for an ideally elastic body, however, a pure tension does not produce a pure yss strain y components exist which constnct the body in the y and z directions,... 

The loss of aromatic character suffered by the thirteen-membered sulfide on passing from the di-trans arrangement of 87b to the mono-trans of 88b obviously results from the inability of the latter to adopt a planar geometry and may best be described as a situation where AH0a becomes dominant in Eq. (1) because of a large angle-strain component. 

The rate of strain components are obtained from Eqs. 6.3-16 and 6.3-17 (simplified for shallow channels)... 

The strain component S12 is usually the deformation of the body along axis 1, due to a force along axis 2 the strain tensor s is usually symmetrical, = s and thus, of the nine terms of s, at most six are unique. Both P and s can be represented as ellipsoids of stress and strain, respectively, and can be reduced to a diagonal form (e.g., P j along some preferred orthogonal system of axes, oblique to the laboratory frame or to the frame of the crystal, but characteristic for the solid the transformation to this diagonal form is a... 

Fig. 5.3 Schematic showing the changes in strain rate and elastic/creep strains of the individual constituents that occur during creep of a composite, (a) Strain rate versus time, (b) strain rate versus in situ stress acting in the fibers and matrix. In both plots, the shadowed portions show the elastic strain components, which compensate the creep rate mismatch of the individual phases, such that the total creep rates of the constituents remain equal. The creep mismatch ratio (CMR) is discussed in Section 5.2.4. After Wu and Holmes.31...

The six independent strain components can likewise be given, as a function of stress, in terms of 36 elastic-compliance coefficients, Sy ... 

For evaluation of the energy minimum with respect to constant pressure (i.e. with variable cell dimensions), first we note that we can define the six components of the mechanical pressure acting on the solid, corresponding to the six strain components, defined in equation (13), that is. 

---