Reference Configurations

Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

Werner H-J 1987 Matrix-formulated direct multiconfigurational self-consistent field and multi reference configuration interaction methods Adv. Chem. Phys. 69 1... 

Eq. (15b) for OH + H2 using multi reference configuration interaction wave functions. 

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. 

If the work assumption is made, i.e., if it is assumed that the external work done by surface and body forces on a finite region in the reference configuration of a body undergoing homogeneous closed cycles of deformation is nonnegative, then an inequality may be deduced paralleling (5.37) by arguments essentially the same as those of Section 5.2.4. 

For some purposes, it is convenient to express the constitutive equations for an inelastic material relative to the unrotated spatial configuration, i.e., one which has been stretched by the right stretch tensor U from the reference configuration, but not rotated by the rotation tensor R. The referential constitutive equations of Section 5.4.2 may be translated into unrotated terms, using the relationships given in the Appendix. 

While r is a spatial vector with components relative to the current configuration, F and its inverse are dual tensors, with one index relative to the current configuration and one index relative to the reference configuration. 

Consider a material line element of length dX in the reference configuration. The motion (A.l) carries this line element into the line element of length dx in the current configuration. From (A.l) and (A.Sj)... 

The triple product of three noncolinear line elements in the reference configuration provides a material element of volume dV. Another well-known theorem in tensor analysis provides a relation with the corresponding element of volume dv in the current spatial configuration... 

The polar decomposition (A. 13) implies that the deformation may be viewed as two successive deformations, the first being a pure stretch from the reference configuration into an unrotated configuration, and the second being a... 

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 ... 

In this case, the components of strain Cy relative to the current spatial con-figuation may be regarded as having been shifted by F via (A.20) to components of the same strain relative to the reference configuration. 

It is evident from their definitions that /, and hence d and w depend on the instantaneous rate of deformation of the current configuration. On the other hand, F and hence U and R relate the current configuration to the reference configuration. In order to find relations for d and w in terms of material derivatives of U and R, the material derivative of (A. 13) may be inserted into (A. 10)... 

If a complementary stress tensor S is defined in terms of the vector T acting on the area dA in the reference configuration by 7 = SN, then, from these equations,... 

Reference Configuration Fuel Max. Flame Speed (m/s) Max. Overpressure (bar)... 

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