The elastic compliance matrix 79&.IT = sll% + S12qyy + s13°ZZ + s142xy + s152yz + S16Tzzx
Eyy = s21% + '22Oyy + '23Ozz + s%zy + sz5zyz + s26Tzx
Ezz = s31% + s320yy + '33% + s34zxy + s35zyz + s36zzx
&"y = s41% + '42Oyy + s43°zz + s44zxy + s45zyz + s4f5zzx
Ey' = s5l% + '52Oyy + s53%. + s542xy + s552yz + s56zzx
&zx = S610xx + s62°yy + s63°zz + '64%~ + s65zyz + s66zzx.
It is not necessary to write these equations in full. An accepted
convention is to use matrix notation, so that the expressions above can be
alternatively written in the abbreviated form
where [E] = and [(TI = and [SI =
The [SI matrix shown above is known as the compliance matrix. In
general, the higher the magnitude of a specific element in this matrix, the
greater will be the contribution to the strain, representing an increasingly
compliant material. 'Compliance' is a form of 'flexibility', and is the inverse
of 'stiffness'.
The compliance matrix is a 6 x 6 matrix containing 36 elements. However,
through considerations of conservation of energy it can be shown that the
matrix is symmetrical. Therefore, in the context of our original assumption
that each strain component is a linear combination of the six stress
components, we find that we need 21 independent elastic constants to
completely characterize a material that follows the generalized Hooke's
law. In the general case, with all the constants being non-zero and of
different values, the material will be completely anisotropic. It is necessary,
particularly for practical applications of the stress-strain relations, to
consider to what extent we can reduce the number of non-zero elements
of the matrix. In other words, how many elements of the compliance matrix
are actually necessary to characterize a particular material?
The key to this study is the architecture of the compliance matrix, and
especially the off-diagonal terms, which have already been emphasized.
For typical engineering materials, there will be non-zero terms along the
leading diagonal because longitudinal stresses must lead to longitudinal
strains and shear stresses must lead to shear strains. The isotropy of the
material is directly specified by the interaction terms, i.e. whether a normal
or shear strain may result from a shear or normal stress, respectively. This
is illustrated conceptually in Fig. 5.10.