Stiffness coefficients 282 matrix

Simplifying the stiffness coefficient matrix by assuming a material with cubic symmetry allows the various bulk elastic properties to be expressed in terms of the three elastic coefficients and various interrelations among them may be derived as illustrated in the following examples. 

Upon inspection of Table 10.3, it can be seen that there are twelve nonzero-valued elastic-stiffness coefficients. Some of these are related by the crystal class and some by transpose symmetry, with the result that there are only six independent coefficients Cn = C22 C12 C13 = C23 C33 C44 = C55 Cee- All other components are zero-valued. Hence, the matrix with all the nonzero independent coefficients designated as such is straightforwardly written as ... 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

At manufacturing level, chemical reactions between the matrix and the liber produce an interface zone of different mechanical properties from the two phases producing it [158]. The load of a composite is usually transferred through the interface between the matrix and the fiber, and the toughness of the composite is determined. Karpur et al. measured ultrasonically the shear stiffness coefficient of the interface in fiber reinforced metal matrix and ceramic matrix composites [158]. They claim that the significance of the quantification of the shear stiffness coefficient of the interface is that the clastic property of the interface can be used as a basis for composite life prediction. 

Other elements of the stiffness matrix are obtained similarly. Apply a unit displacement u, = 1 while Uj =0, j i. Identify the resulting elastic forces and by statics obtain the stiffness coefficients ky. 

The coefficients of the stiffness matrix corresponding to these DOFs are computed following Example 9.4. For instance, to obtain the first column of the stiffness matrix, apply a unit displacement u =l while the other displacements are zero, i.e., Uj = 0,j- 2, 3, , 6 identify the resulting elastic forces, and by statics obtain the stiffness coefficients ... 

The beam stiffness coefficients on the principal diagonal of the matrix P associated with extension, Pn, bending, P55, Pee, and warping, P77, depend on the shell stiffnesses associated with the lengthwise extension, An(s), bending, 5ii(s), and the coupling Bn(s) ... 

With the characteristic ratios obtained above, it may now be possible to check how large the deviations tolerated by Remark 10.8 are. The shell stiffness coefficients All and A33 can be determined by substitution of Eqs. (10.23) and subsequently Eq. (10.22) into those beam stiffness coefficients of Eqs. (10.16), which are exactly kept and contain the respective shell stiffness coefficients to be solved for. Thus, all diagonal entries of the beam stiffness matrix can be calculated as given in Table 10.3 together with the deviations from the example application data, see Table B.l. The great discrepancy in the tensional stiffness Pn is due to the fact that the thin-walled box beam is not able to employ fibers close to its neutral axis, where they would primarily contribute to the tensional stiffness. 

Cap to the Pile Group The resultant pile cap stiffness obtained from step 4 can be added to the diagonal lateral translational stiffness coefficients in the pile group stiffness matrix for the total pile group-pile cap stiffness matrix. 

The parameter AP accounts for a specific contribution of the plastic material to the diffusion process. Phenomenologically speaking AP has the role of a conductance of the polymer matrix towards the diffusion of the migrant (Chapter 6). Higher values of AP in such polymers as PE lead to increased DP-values while in stiff chain polymers such as polyesters and polystyrenes lower AP-values account for smaller diffusion coefficients for the same migrant. The parameters b and c account for the specific contributions of the migrant and the diffusion activation energy respectively. 

These simplifications reduce the size of the elasticity tensors from [9 x 9] to [6 x 6], with 36 elastic coefficients. The shorthand notation normally used for the elasticity tensors are now introduced, namely, that the subscripts become 1 11 2 22 3 33 4 23, 32 5 31, 13 and 6 12, 21. With this change, the elastic stiffness tensor may be written in matrix form as ... 

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