80 StrainDirect coupling of all
normal
componentsIndirect coupling of normal Coupling of normal and
shear components
components(~lDirect coupling of all
associated
shear components
Figure 5.10 The architecture of the elastic compliance matrix.As a first approximation, and in relation to Fig. 5.10, let us assume that
there is no coupling between the normal and shear components and that
there is no coupling of shear components in different directions. This
means that all of the elements designated by the symbols with dense cross-
hatching and left-inclined shading in Fig. 5.10 become zero. We know that
the direct relation between a normal strain and a normal stress is given by
11E: this is because the definition of Young’s modulus, E, is the ratio of
normal stress to normal strain. Hence, all the elements designated by the
vertical hatching will be reciprocals of Young’s moduli. Following the
definition for Poisson’s ratio given in Section 4.6.2, and recalling that this
parameter links orthogonal contractile and extensile strains (which are
manifested by a negative sign in equations containing Poisson’s ratio), all
the elements designated by the wide-cross-hatching will be Poisson’s ratios,
v, divided by a Young’s modulus. Finally, the elements designated in Fig.
5.10 by the right-inclined shading, being the ratio of shear strain to shear
stress, will be the reciprocals of the shear moduli, G.
This results in the reduced elastic compliance matrix shown below:
A material characterized by this compliance matrix has nine independent
elastic constants and is known as an orthotropic material. The nine material
properties are the three Young’s moduli, the three shear moduli and the
three Poisson’s ratios, i.e.