Invariant line strain

ILS invariant line strain LSDA local spin-density approximation... 

Since S is also a shear (but in the opposite direction), its combination with the IPS P produces an invariant line strain (Bowles and Mackenzie, 1954). That is, the combined rotation and Bain strain RB are also the identical invariant line strain (ILS). The ILS reduces to an eigenvalue problem with input data ... 

The time Invariant load-strain curves appear to be an inherent property of the tissue. The values were independent of the porosity of the platens and the strain rates that the tissue was compressed at. Up to about 30% compression the data could be fitted, by linear regression, to a straight line. As such, the curve represents the equilibrium load-strain relationship of the swollen matrix. It should be pointed out, however, that while the strains calculated for this and the other curves are probably close to the actual strains, they nevertheless are approximations, due to the assumptions of shape of the penetrating tissue and the possible slight compression of the tissue by the rod. 

A square or a parallelogram cannot be transformed, for example, into a trapezium as this would contravene the invariance of parallelism. All triangles are, however, equivalent no parallel lines are involved, and successive transformations of shear and strain, in addition to the rigid transformations, will transform any given triangle into any other triangle. 

Fig. 24. Magnetic field dependence of the three C44 modes for T= 4.3 K in CeAlj. Full lines are strain susceptibilities for ( , k ) and (Uj, k ) modes. Broken lines indicate rota-tionally invariant magnetoelastic interaction (Liithi and Lingner 1979).

Figure 1. The remanent strain saturation curve dividing remanent strain space into regions that are attainable and unattainable by a polycrystal assembled from randomly oriented tetragonal single crystals. Only remanent strain states below the curve are attainable by such a material. The dots are numerical results from Landis (2003a) obtained using a micromechanical self consistent model, and the line is one divided by the function/given in Eqs. (2.7) and (2.8). The remanent strain invariants 4 and are defined in Eq. (2.5) and the results are normalized by the saturation strain in axisymmetric compression s. ...

The quality of the approximation deteriorates when significant strains are present in the structure (like those in ice nanotubes). Figure 3 shows the correlation between the calculated energy and its graph invariant approximation for clusters (1) and (5). The former is not strained, while the latter has mild strains. In line with this observation, the quality of the approximation is better for stable INT and INTg than for strained INT and INT . As for the indices m, the value of depends on its parity is larger for... 

We recall from the discussion in Sec. 2.3.5 that a necessary condition for the appearance of a polarization was that the medium lacks a center of symmetry. The reason for this was that since at equilibrium the stress as well as the strain will be centro-symmetric, the piece of matter cannot develop charges of opposite sign at opposite ends of a line through its center if it has a center of symmetry, in accordance with Curie s principle. On the other hand, inspection of Fig. 31 immediately reveals that none of the three strains splay, twist, and bend has a center of synametry. Hence Curie s principle allows a local polarization to appear as a result of such local deformations in the director field, even if the medium itself has a center of symmetry. We see from Fig. 31 that the splay deformation violates the ->- invariance, whereas twist and bend... 

To determine the tangent modulus, a straight line is drawn tangent to the curve the slope of which is reported. Typically, the steepest part of the eurve is selected this is invariably the initial slope, in which case it is reported as the initial modulus. When the tangent is taken at a specific strain level, this value must also be reported. To determine the chord modulus, two points on the eurve are selected and the slope of the line connecting them is calculated. When the initial point chosen is 0% strain, the value is reported as the secant modulus of the upper level, e.g., 1% or 2% secant modulus. In all modulus calculations it is assumed that deformation is homogeneous at low strains and that deformation... 

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