Compliance constants

Swanson, B. I. and S. K. Satija. 1977. Molecular vibrations and reaction pathways. Minimum energy coordinates and compliance constants for some tetrahedral and octahedral complexes. J. Am. Chem. Soc. 99 987-991. 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

The coefficients Cn are called elasticity constants and the coefficients Su elastic compliance constants (Azaroff, 1960). Generally, they are described jointly as elasticity constants and constitute a set of strictly defined, in the physical sense, quantities relating to crystal structure. Their experimental determination is impossible in principle, since Cu = (doildefei, where / i, and hence it would be necessary to keep all e constant, except et. It is easier to satisfy the necessary conditions for determining Young s modulus E, when all but one normal stresses are constant, since... 

The elastic constants derived by Van Fo Fy and Savin are as follows. (The symmetry axis is 3, c is the concentration of the circular reinforcing phase in a hexagonal array. The compliance constants Sy are quoted)... 

The associated hardness (compliance constant) is defined by a similar minimum energy constraint ... 

The constants s and c ( = 1 /s) are known as the elastic compliance constant and the elastic stiffness constant, respectively. The elastic stiffness constant is the elastic modulus, which is seen to be the ratio of stress to strain. In the case of normal stress-normal strain (Fig. 10.3a) the ratio is the Young s modulus, whereas for shear stress-shear strain the ratio is called the rigidity, or shear, modulus (Fig. 10.36). The Young s modulus and rigidity modulus are the slopes of the stress-strain curves and for nonHookean bodies they may be defined alternatively as da-/ds. They are requited to be positive quantities. Note that the higher the strain, for a given stress, the lower the modulus. 

If, however, one assumes uniform stress throughout the same nontextured polycrystal a similar averaging procedure can be performed over the elastic-compliance tensor using the corresponding nine elastic compliance constants Sn, S12, S33, S44, S55, Sss, S12, S23, and S31. This is known as the Reuss approximation (Reuss, 1929), after Endre Reuss (1900-1968), and it yields the theoretical minimum of the elastic modulus. 

Similarly, from Table 10.4, there are six independent elastic-compliance constants Sii,Si2,Si3,533,544, and See. Substitution of these relations in Eq. 10.27, and using the relations given in Eqs. 10.28-10.30, gives ... 

Assuming (1) axis is the longitudinal direction of the fibre, the compliance tensor 5c takes the same form than Si, whereas, taking into account the transversal isotropy, the compliance constants have the following expressions ... 

Strong Evidence for an Unconventional 1,2-(C—>P)-Silyl Migration DFT Structures and Bond Strengths (Compliance Constants)... 

Table 3. Lengths and compliance constants (COCO [AVaJ]) of selected bonds. Dark grey C-Si bonds involved in the transition state (cf. text).

Relative strengths of selected bonds in the course of the 1,2-silyl migration as measured by compliance constants in [AVaJ]. Compliance constants are inversely proportional to bond strengths. 

Here Ctju are the stilfness constants and Sijki are the compliance constants. They form two symmetric fourth-rank tensors with 81 elements inverse one to another. For the triclinic symmetry only 21 elements are independent because the strain and stress tensors are symmetric. Consequently the indices i,j and k, I can be permuted and also can be permuted one pair with another. For a crystal symmetry higher than triclinic the number of independent elastic constants is less than 21. 

Table 12.5 The matrix C of the stiffness constants for all Lane classes. The matrix S of the compliance constants is identical, only C is replaced by S. The last column gives the number of the independent constants.

The index V was added to distinguish these constants from the stiffness constants obtained by inverting the tensor of averaged compliance constants, calculated by integrating Equation (74b) over the Euler space. 

Here Sy are the averaged compliance constants for the isotropic polycrystals. They are calculated from the single-crystal compliances with formulae similar to Equations (89) ... 

In this expression, similar to Equation (84a), the first term is the strain of the isotropic matrix given by Equation (94). The second term is the strain induced in crystallite by the matrix and is given by the Eshelby" theory for an ellipsoidal inclusion. The tensor lifg) accounts for the differences between the compliances of the inclusion and of the matrix and has the property ty = 0. To calculate the peak shift. Equation (105) is replaced in Equation (67b), which is further replaced in Equation (83). Analytical calculations can be performed only for a spherical crystalline inclusion that has a cubic symmetry. For the peak shift an expression similar to Equation (91) is obtained but with different compliances. According to Bollenrath el the compliance constants in Equation (91) must be replaced as follows ... 

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