Matrix compliance

In order to describe completely the state of triaxial (as opposed to biaxial) stress in an anisotropic material, the compliance matrix will have 36 terms. The reader is referred to the more advanced composites texts listed in the Bibliography if these more complex states of stress are of interest. It is conventional to be consistent and use the terminology of the more general analysis even when one is considering the simpler plane stress situation. Hence, the compliance matrix [5] has the terms... 

By a similar analysis it may be shown that, for applied stresses (rather than applied strains) in the global directions, the overall compliance matrix [5] has the form... 

In order to get the strains in the global directions it is necessary to determine the overall compliance matrix [5]. This is obtained as indicated above, ie... 

Having obtained all the terms for the extensional stiffness matrix [A], this may then be inverted to give the compliance matrix [a]. 

The laminate properties may then be obtained as above from inspection of the compliance matrix. 

Overall Stiffness Matrix Overall Compliance Matrix... 

As [fi] = 0 in this symmetric laminate, the compliance matrix is obtained by inverting [A]... 

Most simple materia characterization tests are perfomned with a known load or stress. The resulting displacement or strain is then measured. The engineering constants are generally the slope of a stress-strain curve (e.g., E = o/e) or the slope of a strain-strain curve (e.g., v = -ey/ej5 for Ox = a and all other stresses are zero). Thus, the components of the compliance (Sy) matrix are determined more directly than those of the stiffness (Cy) matrix. For an orthotropic material, the compliance matrix components in terms of the engineering constants are... 

This mathematical condition can be replaced by the following physical argument. If only one normal stress is applied at a time, the corresponding strain is determined by the diagonal elements of the compliance matrix. Thus, those elements must be positive, that is,... 

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

The generalized compliance matrix, combining the system electronic and geometric degrees-of-freedom,... 

One can explicitly express the compliance matrix in terms of the elements of the principal charge sensitivities defining the generalized electronic-nuclear hardness matrix H of Equation 30.8, by eliminating AN and AQ from Equation 30.9 ... 

Eliminating AF from the second Equation 30.14 and inserting the result into the first of these two equations give the following transformation of the representation independent perturbations (A/a, AQ) into the linear responses of their conjugates (—AN, —AF), expressed in terms of the matrix elements of the compliance matrix S of Equation 30.12 ... 

The fully inverted compliance matrix, S(,4—B) = H(,4—B) which determines... 

Finally, a possible use of these coupling constants as reactivity indices has been commented upon in both the one- and two-reactant approaches. In the interreactant decoupled applications the molecular compliants, obtained from calculations on separate reactants, can be used directly to qualitatively predict the intrareactant effects resulting from the interreactant CT. The building blocks of the combined electronic-nuclear Hessian for the two-reactant system have been discussed. The corresponding blocks of the generalized compliance matrix have also been identified. In such a complete, two-reactant treatment of reactants in the combined system, the additional calculations on the reactive system as a whole would be required. 

Decius, J. C. 1963. Compliance matrix and molecular vibrations. J. Chem. Phys. 38 241-248. Dreizler R. M. and E. K. U. Gross. 1990. Density Functional Theory An Approach to the Quantum Many-Body Problem. Berlin Springer-Verlag. 

Equation (5.7) related the stress tensor to the strains throngh the modulus matrix, [Q, also known as the reduced stiffness matrix. Oftentimes, it is more convenient to relate the strains to the stresses through the inverse of the rednced stiffness matrix, [2] called the compliance matrix, [5]. Recall from Section 5.3.1.3 that we introduced a quantity called the compliance, which was proportional to the inverse of the modulus. The matrix is the more generalized version of that compliance. The following relationship then holds between the strain and stress tensors ... 

Finally, the time-dependent stiffnesses in Equation 8.40 can then be obtained through inversion of the compliance matrix following the quasielastic approximation ... 

The elastic midplane strains and curvatures are related to the resultant forces (AT,) and moments (AQ through the laminate compliance matrix... 

Because Nt = 0 and Mx = — M2 during cure, the pertinent compliances for cross-ply laminates are Su and S12. The StJ compliance matrix can be found from [19]... 

The factors in (6.27) take care of the fact that two terms such as sxy + eyx are only counted once in the abbreviated notation. In this notation the compliance matrix sij is the inverse of the stiffness matrix Cj/. 

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